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Now, I doubt there are many sources talking about the feelings of Union soldiers about this song specifically. There must be some letters where soldiers talked about the popular tune, but I don't think there are whole studies. Instead, I'll focus on two things: reaction to the Harpers Ferry raid, and abolitionist sentiments within the Union Army, which can give us an idea of how the average Northern soldier felt about a war for abolition, and by extension about a song that so explicitly endorsed abolition.

I would not characterize the reaction as completely negative, at least among the majority of Northerners that would go on to vote for Lincoln in 1860. Sure, it was not completely positive either, but Brown became a hero throughout the North as a result of his actions.

It's important to remember the idea of the "Slave Power". Republicans, including Lincoln and Seward, believed that the Slave states and their Northern allies had joined together in a big conspiracy to subvert the government and use it to protect and spread slavery. They saw this as an existential threat. The idea that the nation could not remain half-slave and half-free, articulated in the Lincoln's "House Divided" and Seward's "Irrepressible Conflict" speeches, meant that sooner of later either the Free Labor system of the North or the Slave economy system of the South would triumph. Since they saw Northern society as the happiest, freest and most perfect system, and the South as the antithesis for everything they cherished, the idea that a Southern cabal was trying to force hated slavery upon them was terrifying.

When you look at the events of the 1850's, the idea seems plausible. The Fugitive Slave act had brought slave catchers to the North and forced Northerners to help them. The Kansas-Nebraska act had sullied the territory with slavery, and then Southern ruffians had used murder and terror to try and force slavery them against the wishes of most inhabitants. President Buchanan had helped them in this bid, and he had apparently conspired with the Supreme Court to extend slavery to the territories with the Dred Scott decision.

For most northerners this was no simple political game. It was an existential battle for whether slavery or freedom would become the guiding principle of the US, and the Slave Power was winning. They felt humiliated, they felt angry, they wanted change. And then came a hero who actually did something, who dared to strike at the hated Slavocrats. That hero was John Brown.

The first reaction to the raid was, to use McPherson's words, "baffled reproach" They called him a misguided madman. Garrison, for example, said that Brown's raid "though disinterested and well intended" was "misguided, wild, and apparently insane." But Northern opinion soon started to change. Brown consciously cultivated an image as a martyr for liberty during his trial. The dignity and fortitude he showed during it impressed even some Southern Fire Eaters - you can only imagine the impression it made upon Northerners!

Many abolitionist were so because they thought slavery was a sin, a monstrous injustice condemned by God himself. That's why works such as Uncle Tom's Cabin produced such a deep reaction within Northern hearts, because they tapped into the religious fervor that laid within them. Even in those that were not that religious, the Christ-like portrayal of Uncle Tom and its appeal to Christianity was enough to awake a strong opposition to slavery. You can even see this in the strong religious undercurrent in the Battle Hymn of the Republic, and its famous final line "As He died to make men holy, let us die to make men free". Brown knew this, and framed his raid as an attempt to right a great wrong, an attempt for which he was now being martyred. He told his wife that he could recoup all lost capital "by hanging a few minutes from the neck" and later said that he was "inconceivably worth more to hang than for any other purpose."

Then came his execution, and the lines that moved thousands of Northerns, and can move people even now:

I deny everything but what I have all along admitted: of a design on my part to free slaves. . . . Had I interfered in the manner which I admit . . . in behalf of the rich, the powerful, the intelligent, the so called great . . . every man in this Court would have deemed it an act worthy of reward rather than punishment.

This Court acknowledges, too, as I suppose, the validity of the law of God. I see a book kissed, which I suppose to be the Bible, or at least the New Testament, which teaches me that all things whatsoever I would that men should do to me, I should do even so to them. It teaches me, further, to remember them that are in bonds as bound with them. I endeavored to act up to that instruction. . . . Now, if it is deemed necessary that I should forfeit my life for the furtherance of the ends of justice, and mingle my blood further with the blood of my children and with the blood of millions in this slave country whose rights are disregarded by wicked, cruel, and unjust enactments, I say, let it be done.

The effects of this words were electric. Theodore Parker said that Brown was "not only a martyr, but a Saint", while Ralph Waldo Emmerson asserted that the old warrior "would make the gallows as glorious as the cross." Henry David Thoreau compared his execution with that of Jesus Christ, and in the wake of it Garrison, normally a pacifist, wished "success to every slave insurrection at the South and in every slave country."

It was not only these intellectuals. Thousands wept at the casket of old John Brown. McPherson can describe the events better than I ever could.

Extraordinary events took place in many northern communities on the day of Brown's execution. Church bells tolled; minute guns fired solemn salutes; ministers preached sermons of commemoration; thousands bowed in silent reverence for the martyr to liberty. "I have seen nothing like it," wrote Charles Eliot Norton of Harvard. More than a thousand miles away in Lawrence, Kansas, the editor of the Republican wrote that "the death of no man in America has ever produced so profound a sensation. A feeling of deep and sorrowful indignation seems to possess the masses." A clergyman in Roxbury, Massachusetts, declared that Brown had made the word Treason "holy in the American language"; young William Dean Howells said that "Brown has become an idea, a thousand times purer and better and loftier than the Republican idea"; Henry David Thoreau pronounced Brown "a crucified hero".​

Brown had transcended and become an ideal, the symbol of the fight against the slave power that had dared to humiliate and threatened the North year after year. Conservatives in vain tried to assure the South. Even Republican attempts to distance themselves from Brown carried an undertone that they disapproved of his methods, not his cause. The South was outraged. It's no exaggeration, in my opinion, to call the John Brown raid the moment where the Civil War became truly inevitable. Brown was a hero for Northerners who dreamed of destroying the Slave Power, and when that Slave Power rose in rebellion, many looked to him for inspiration as the first of the brave warriors that was now going to crush treason.

It's true that they were not abolitionist, but for most of them John Brown did not represent abolitionism, but rather finally standing up to the Slave Power and fighting back after a decade of humiliation. This is more apparent in the original and simpler version that talks about him as a warrior and says that they will "hang Jeff Davis to a sour apple tree", but doesn't really talk about slavery. In the original version, the message is that of a warrior who strikes back against treason and rebellion - this is what Brown represented to them.

It is true, however, that Brown was also the icon of abolitionism, thus the more explicitly abolitionist message of other versions. That begs the question, was the average Northern soldier an abolitionist?

"John Brown's Body" was the unofficial anthem of the North, just as much as "Dixie" was that of the South, an irony that many will note as "Dixie" was written by a northerner, and "John Brown's Body" generally being thought to have been set to an existing Southern folk-tune, although there is more debate on the specifics.

Anyways though, to stay on topic, the claim that it originated in the 12th Mass. to make fun of a soldier literally named John Brown is not entirely accepted, even if many agree with it but it is almost besides the point, as the origin story that it related to Sgt. John Brown is based on the playful ribbing his fellow soldiers gave him for the name, such as "This can't be John Brown - why John Brown is dead!" as well as the fact he was a short, stocky man whose knapsack looked amusing on the frame. Soon, the Battalion's glee-club had written up lyrics to accompany the tune of "Say Brothers Will You Meet Us on Canaan's Happy Shore", and it became an instant hit with the unit, expanding to the whole regiment which became known as "The Psalm singing regiment from Boston".

It perhaps isn't a coincidence that the song originated in Boston, which was generally seen as more of an Abolitionist stronghold than, say, New York, but I also would be careful not to overrate the strength of Abolitionist sentiment in the original incarnation, as only one of the lines you quote, "He is gone to be a solider of the Lord", can actually be attributed to that first version that originated with the 12th Mass. Simply put, it was not an Abolitionist song as originally conceived, and as it moved beyond its origins and others missed the joke, commentators were quite struck by the fact John Brown was being touted in song. A reporter, seeing a Massachusetts unit parading down Broadway on their way South, noted:

Who would have dreamed that a thousand men in the streets of New York would be heard singing reverently and enthusiastically in praise of John Brown.

This is fairly important. One didn't need to be particularly Abolitionist to be attached to a song about John Brown. It was a catchy tune, and at least originally for the soldiers, quite light-hearted. Even moving beyond the joke though, that sentiment wasn't necessary. Brown's execution in 1859 had seen a significant Northern celebration of him as a martyr, and this had of course royally pissed off the South. All that a Billy Yank really needed to feel to enjoy the song was that by doing so, it might needle ol'Johnny Reb a bit. The song was said to have only truly come into its own as the song of the Union soldier in 1862 outside Yorktown when some men began to sing it while under Confederate bombardment, and after which it took off within the Army of the Potomac. It both bucked up their own spirits, but also was a "special taunt" that invoked "the name of the grim old Moloch, whom - more than any one, Virginia hates." Perhaps more than anything, knowing that it annoyed the Confederates enshrined its appeal for the boys in blue.

That didn't change the fact that many saw the mere evoking of the name as being in incredibly poor taste. Especially northern Democrats, but some Republicans as well, as Abolitionism was hardly synonymous with the party and Brown's choice of violence still controversial, dismissed with various epithets ranging from simply "hideous" to "negro doggerel". As it grew in popularity and came to be sung in schools, concerns about the song and exposure to children saw attempts to remove it, although the fights generally just proved to be nothing more than political factionalism on school boards.

Even that wasn't always a barrier though. Although some Union troops recall verses being answered purposefully by a barrage of Confederate cannon, even the greycoats were known to at times sing the song, certainly without any strongly Abolitionist sentiments in the verses, but they too would sometimes belt out the original classic that "John Brown's Body lies a-mold'ring in the grave [x3]; His soul is marching on." And of course while you note that some verses were added, this undersells just how varied the song could be, with plenty of variation, either solo or just new verses, that reflected the soldiers' lot.

When McClellan returned to command for instance, a version heard was:

Brave McClellan is our Leader now x3
With him we're marching on!

And others reflected war aims different than Abolition, such as the following verse which was one of the first added to the original to the point that it is almost canonical:

We'll hang Jeff Davis on a sour apple tree (x3)
As we go marching on.

And German-Americans even came up with their own version to march to, which evoked the failed Revolutions if '48 that had driven some of them to America:

Wid sind Deutsche und wir kaempfen
Fuer die Freiheit der Union
Fest im Glauben an die Einheit
So wie "48" schon
Yankee-Doodle auf den Lippen
1st Gerechtigkeit der Lohn
Fuer das Banner der Union!

Auf, fuer Lincoln und die Freiheit. (x3)
Fuer das Banner der Union!

As did African-American troops who wrote up versions that often revolved around their desire to make a contribution to the war, such as:

Oh! We're the bully soldiers of the "First ob Arkansas,"
We are fightin' for the Union, we are fightin' for the law,
We can hit a Rebel further then a white man ever saw,
As we go marchin' on.

While some versions still might include sentiments that referenced freedom or emancipation, the most fervently abolitionist versions generally weren't always coming from the soldiers. Hearing the song sung by the soldiers, Abolitionists were inspired to build off it, if not entirely transform it. "Battle Hymn of the Republic" of course was a full rewrite by Julia Ward Howe, but other Abolitionists just did additional verses, such as Edna Dean Proctor who wrote up:

John Brown died on a scaffold for the slave;
Dark was the hour when he dug his hallowed grave;
No God avenges the life he gladly gave-
Freedom reigns today!

Just as black soldiers had embraced the tune, African-American civilians also did, generally with a more Emancipationist bent. One version heard, sung as enslaved persons in DC waited for the clock to strike midnight and their liberation to come, cried out:

The first of January next, eighteen sixty-three
So says the Proclamation, the slaves will all be free! 
To every kindly heart ’twill be the day of jubilee;
For the bond shall all go free! 

The interesting thing though is that while the two essentially started out separately, they did converge. Although a small core at the beginning, abolitionist sentiment within the Union Army grew through the war, and at least some saw more than mere coincidence, but that the attachment to the song had actually helped that change in sentiment by the soldiers. Perhaps it goes too far, and it is better to say that it was growing belief that destroying slavery was central to their mission that helped "John Brown's Body" grow in popularity rather than the reverse, but certainly it can't be said that there isn't a smidge of truth, with at the very least both helping each other along.