And were any Union equivalents to the Confederate Home Guard militias given the task to look for these soldiers post-war?
The short answer is no.
After the Civil War ended, the United States was a nation of concilliation, to a point. The paroles and terms of surrender that U.S. Grant and R.E. Lee agreed upon are cited for allowing there to be no prosecution of Confederate parolees as long as they abided by their terms of surrender. Any parolled northern man who took up arms with the Confederacy would have been subject to the same terms. And these terms, similar to the ones agreed upon by Grant and Lee, were used by other surrendering armies as the war was snuffed out.
Prosecuting northern men for taking up arms in the Confederate Army, for the sole reason that they were from northern states, would have been immoral. I say this because if you aren't prosecuting men from southern states, which they didn't, then what ground would you have to prosecute someone from the north? The soldiers from the southern states who fought for the Confederacy were just as American as the men of the northern states who fought for the Confederacy. Everyone was an American.
And if I can posit my own theory, I'd say that a policy of prosecution would have had dire ramifications for the country; you'd drive a wedge between the people you were trying to reunite.
When I think of northern men who fought for the Confederacy, John C. Pemberton always comes to mind. He was from Pennsylvania; his wife was born in Virginia, and like so many other officers who fought for the south, he resigned his commission in the Union Army at the outset of the war in 1861.
He quickly rose to the rank of Lt. General in the Confederate Army, and famously opposed U.S. Grant during the Vicksburg campaign. After the fall of Vicksburg, he was relegated to serving as a Lt. Col in the artillery. Quite a fall from the perch of General, if you ask me. I can't think of another case where a General went back to serving as a Lt. Col during the Civil War. When Pemberton was captured late in the war he was simply paroled. This is a good representation for all of the northern men who were parolled after serving in the Confederate Army.
To clarify, are you asking if Northerners who fought for the South were prosecuted post-war?
Perhaps the most high profile wouldve been LTG John Pemberton. He was the commander of the Confederate Army defending Vicksburg and he consequently made the decision to surrender on July 3/4 1863 to Grant after a few weeks of siege.
He was actually paroled and exchanged and returned to Richmond and expressed a desire to his friend Jefferson Davis to return to active service. However giving him any meaningful post had become politically impossible within the army. He did though serve as an artillery commander on the siege lines around Richmond. He was captured in North Carolina at the end but like the rest of the Confederate Army was allowed to give his parole and live unmolested. His nephew was actually the John Pemberton who invented Coca-Cola!