I can't address your question specifically on the 3rd Missouri directly (though it's fascinating to hear about given what I'm about to say), but I can absolutely say that there were many leftists/european republicans/proto-socialists/etc fighting for the Union in the American Civil War. Probably among the most famous of them was August Willich, a Prussian noble by birth who gave up his landed title and was one of the first true Communists, even being mentioned and debated by Karl Marx in many of his dispatches. He served in the Union army, reaching the rank of Brigadier General. He fought in the western theater, as a matter of fact his men were some of the first to charge up Missionary Ridge at Chattanooga. Earlier, at Shiloh, when he was a colonel, he rallied his regiment (an Indiana regiment if I recall correctly, though I don't recall the number) by having the regimental band play La Marseillaise.

I'd be very interested in hearing if anyone else knows about the 3rd Missouri, though, and I'm definitely gonna do some of my own digging now.