Given how expensive repeating arms like the Henry and even Spencer rifle were, was it even feasible for the North to have adopted anything but the Springfield Rifle Musket during the Civil War? I'm already aware of Stanton's(?) decision to turn down breechloading and repeating arms because of concerns over ammunition supply, but I haven't been able to nail down a concrete answer over whether the North could actually have kept large units supplied with enough ammunition to allow more rapid fire. As far as I know, the North already had significant challenges funding the war and supplying troops in the field, and potentially increasing ammunition consumption by 300% doesn't seem like something that could be easily done.
So this isn't directly focused on your question, but this older answer of mine does touch on it as part of the larger development of breechloading firearms. The key point to your question, as mentioned there, is that shifting production on large enough a scale would have been incredibly disruptive, so while arming on a unit-by-unit basis from a wide variety of small manufacturers was quite possible, an entire overhaul of the Union Army's armament would have been a very different, and near insurmountable. There were other more basic issues, as mentioned by /u/Bodark43 below, but at the end of the day, even if the Army brass had been gung-ho, a full turnover would have been nearly impossible.
There had already been some use of breechloading weapons before the Civil War. John Hall had made his breechloading rifles in a shop annex at the Harper's Ferry Armory, in the 1830's, and the Navy had tried the Jenks rifle in the 1840's. These designs were not perfect- the Hall leaked enough gas between the breech block and barrel to seriously hurt fingers, if they were not kept out of the way, and the Jenks could get clogged with fouling. But though there was some resistance to the idea of breechloading guns by the officer corps, the great success of armies using breechloading weapons against muzzleloaders in Europe, and successful experiments by the War Dept pretty much settled the question. By 1859, the Secretary of War would write, " These arms commend themselves very strongly for their great range and accuracy of fire at long distances; for the rapidity with which they can be fired....With the best breech loading arm, one skillful man would be equal to two, possibly three, armed with the ordinary muzzle-loading gun." So, though there were fears by the officers that soldiers would simply shoot off all their ammunition, there were no worries expressed about paying for this, by the War Dept. They sensibly saw this as arithmetic: why pay for three soldiers with three guns, when you could pay for one soldier shooting as fast as three?
However, there was concern about the great variety of guns needing to be supplied with a corresponding variety of ammunition. There was the Spencer, the Sharps, the Smith, the Merrill, the Colt, the Joslyn, the Gwyn & Campbell...there was even the Burnside, designed and put forward by General Ambrose Burnside. The Spencer, Henry, Burnside and Smith had metallic cartridges that were specific to those guns ( the Smith actually had a rubber one...but that's beside the point..)and there were complaints about the expense and difficulty of keeping track of them all. There had been famous mishaps that came from supplying armies with different guns ( the British soldiers sorely beset defending Les Haies at Waterloo were crushed to find that the valiant crew bringing them new supplies of ammunition had brought the wrong stuff, for example.) and so there were understandable concerns about getting the right stuff to the right units- and some griping about the higher cost of metallic cartridges over paper cartridges ( and the Henry got poor reviews , for expense). But this was because no one single design had been adopted yet- that would have to wait until 1866.