This discussion has been going on for fully 3 weeks, with no "solid" resolution of it. So, although I do not know for certain what the item's ID is, I think I should contribute my thoughts and observations, in the hope that they might lead to some additional progress toward the item's correct ID.
I notice that the item's face appears to actually be slightly concave ...like a old-style dinnerplate. That made me think of an overalls fastener-stud -- but at the reported size of a US quarter-dollar (.95-inch diameter), it's probably too big to be that. Nonetheless, because its back is torn-open in the way it is, I'm not sure it had a typical button-loop on it.
The item's face needs additional cleaning. Seeing more details of the bird could provide more-definite ID clues. In its current encrusted condition, to me the bird doesn't quite look like a typical American eagle-symbol. For one thing, its feet don't look like those of an eagle.
Also, in its current encrusted appearance, the bird's neck looks much longer than we see on a typical American eagle-symbol ...but that might just be due to the crud between bird's neck and the wing-tops. Currently, its feet and longer neck remind me of a turkey. Cleaning the crud off might show us what we need to see.
Like Ivan, it occured to me when I first saw this item that the 30 stars might be date-significant ...because the number of stars is quite significant in dating US flags ...especially civil war era flags.
I may be seeing a bit of silver-plating (or nickel-plating, or perhaps gold) in the depressed area around the stars -- but I'm not certain about that, due to the heavy encrustaion. Again... more cleaning, please.
If it does have some plating, knowing whether it is silver or nickel-plate will help to date the item. Nickel-plate came into favor as a widely-used substitute for silverplate on buttons (and similar items) in the 1870s/80s ...and ever since ...for two reasons: it is significantly cheaper than silver, and it doesn't tarnish or wear off as easily as silver.
Because Celtex lists Texas as his/her location, I assume the item was found in that state ...which could be relevant to the item's ID. The item indeed might not have an American eagle-symbol, but instead, a Mexican version. Right away, along with the bird's odd feet, I noted the absense of the usual American symbol's typical arrows-&-olive-branch. That being said... on Mexico's National Seal emblem, the eagle is gripping a snake in its talons.
That's all I've got about the item, until we get to see it cleaned-up.
For that, I'd suggest a soak in lemon juice, and then the gentle use of a toothpick on the encrusted areas.