That's quite the unusual button! Some of the oddities being the filled back, 12 start and circle orientation over the eagle, the feathers not attached to anything, the hooked beak and crest on the eagle's head, and finally the arrows pointing down as opposed to the opposite on every other example of military button having arrows.
I did do some extensive searching on the net and compared literally hundreds of buttons from the start of 2 piece buttons through WW1 era and there is nothing like it to be found. But there are very many similarities to CW era buttons from both sides of the war as well as a mix of styles from various manufacturers including the odd looking head, height of the wings, profile of the neck, etc. But none actually match any particular style. The Eagle itself looks to be a mix of eras from the early to mid 19th century.
All put together leads me to think it's an oddball button probably confederate made by an amateur maker who wasn't really up on details but instead tried to copy similar variations which would explain the poor artistry and upside down arrows. The 12 stars is unusual as there were 11 states in the confederacy, although many thought that there were actually 13 states, so maybe the maker also got that wrong? As for the filled back, it could be a repair, but as there is an eyelet still there my guess is it was made that way an it would probably be easier than making a 2 piece, possible using lead from a bullet or other available source.
So anyway, that's my 2 cent's worth of guesswork. I would suggest contacting some of the CW museums and get opinions of the experts there. I would probably start with Shiloh Relics or the National Civil War Museum and go from there. It's possible you might have a very rare or unique historical item, and I would be very interested to know what you find. Keep us updated on your research!