Actually, that particular version of eagle-A button is from a lot earlier than the civil war. It's peculiar version of spread-winged eagle, having a skinny neck, long skinny wings, and featherless skinny "shins" above the talons, is one of the 1830s/1840s versions of eagle-button eagle. In general, 1850s and civil war ones had a thicker neck and wings, and feathered fatter shins.
Callicles, I must clarify what I said about eagle-letter buttons being for low-level officers. Before the 1854 advent of the "un-lettered shield" eagle button, just for Enlisted-men's ranks (non-officers), all US Army ranks wore a lettered shield button, except for the Engineer Corps, which had its own distinctive emblem on its buttons.
Grasshopper, congratulations, it's cool to find a version of eagle-A button, especially a 2-piece one, which is not shown in the Albert button-book. ("Record Of American Uniform And Historical Buttons / With Supplement" by Alphaeus H. Albert.) Your button might be shown in the Tice button-book, but the time as I type this is 1:15AM EDST, so checking my Tice book will have to wait until the weekend.