I love finding buttons, but I don't know a whole lot about them. It's a two piece cuff button with an iron back. Here in the PNW we don't have as much military history as they do east of the Mississippi. The thumbtack is for size comparison.
Actually, you are pretty much correct. The US military did have a cuff size button that went on the kepi that had an iron back, but it was against the leather band. The Confederates had what is referred to as locals that often had an iron back. But I haven't been able to find that particular button. The eagle appears to be a type that was used before the war. And it seems to be a very good stamping. But I have no idea where it belongs.