It is a US Army Enlisted-Men's "general service" button. The specific version you found was issued from 1875 to 1902. It is distinguished from the earlier version (issued 1854-1874) by having a high-raised shield on the eagle's chest, whereas the previous version's shield was more-or-less "flush" with the eagle's chest. For visual comparison of those two versions, in the photo below, yours is in the middle of the row, and the 1854-1874 version is on the right.