In 1861, at the outbreak of the American Civil War, Giuseppe Garibaldi volunteered his services to Abraham Lincoln. He was offered a major general's commission, but Garibaldi wanted to be commander-in-chief of the Union armed forces and to have the power to declare the abolition of slavery. These conditions couldn't be met, and so he was not involved in the conflict.
Let's say the American government agreed to Garibaldi's conditions, or that the conditions he requested were less demanding, or that he just accepted the commission he was offered. What might happen?
A lot depends on what exactly he will be asked to do. I can't see Lincoln agreeing to his demands, so he will have to tone them down for this to work. He could be assigned a generalship, certainly not overall command of Union armies. ACW was not the kind of warfare he was accustomed to, albeit no one would know that yet in 1861. He may do well in a Western command, probably less so in Virginia. If he occupies Confederate territory, he would do his best to free slaves in areas under control, pushing the envelope of political instructions from Lincoln as much as he could. This radicalizes the political side of the conflict and puts slavery even more to the forefront, possibly forcing an earlier EP.
Backlash over a foreigner with an important command doing politically sensitive things such as freeing slaves is likely in order, but if he wins battles (and well could) that may be contained. He would be followed by Italian volunteers, probably a few thousand of them. The USA become a destination of choice for disgruntled Italian radicals, republicans, Garibaldine veterans and the like, more so and earlier relative to OTL. Given their willingness to fight for the Union, American views of Italian immigrants are more positive.
If the Union wins the war (as expected) and Garibaldi acquaints himself well in the fight, that would add considerably to his heroic aura. Also, he would not be in Italy to stir up things about Rome, which might delay the September conventions and impact Franco-Italian relations more generally.
Would he stay in the USA post war?
If so, Italian performance in 1866 is even more dismal than IOTL, not that it would make much difference in the endgame unless other butterflies cause French intervention. I think he would go back to Italy though, and maybe receive a more important command. With the experience and prestige acuired in the ACW, he could, and that is bad news for the Austrian troops in Northeast Italy in 1866.
All in all, this slightly increases the odds of a general European war in which France and Austria fight Italy and Prussia (which Russia might possibly join).