Agreeing with nhbenz... your button is definitely what button-collectors call a "Jacksonian" Patriotic-emblem button. They are called Jacksonian because they became quite popular with the American public during the two-term presidency of Andrew Jackson (1829-1837). The eariest ones were made in Britain, but American button-makers soon copied the British ones. Their construction is a flat brass disc with a separate rim crimped around the edges of the flat disc... which is a very early version of what are known as "two-piece" brass buttons. One of the photos I'll post below shows the separate rim has pulled away from the flat button's edge, at about 5:00.
Jacksonian buttons range in size from about 13mm to a maximum of 19mm... they're all "cuff" or "vest" size... none are "coat" size.
Jacksonian buttons are not known to have ever been issued by the US Army. The majority showed a non-military emblem/design on their front. But there are also several varieties of "Patriotic" Jacksonian buttons, showing the spread-winged American eagle. You might find yours somewhere in the photos posted below.