I have been fortunate enough to have found a few, especially since for the military ones, I hunt nowhere near where any action took place.
Two of the buttons with Eagles on them are unique, being the only known examples of them, so they are my favorite button finds of all time.
United States Topographical Engineer button ca. 1820, Albert's TE-1 known as the USTE button

War of 1812 Union Greens button, Irish militia unit from Baltimore Md, which fought in the Battle of North Point saving Ft McHenry during the battle when the National Anthem was written.

Patriotic button, Alberts UU132A

1820 era button with a Naval Eagle backmark

Jacsonian Eagle button

Early US Naval button

Congressional Campaign button of 1834

War of 1812 era button

1840 era Naval button

Several Eagle backmarked buttons
I have other Eagle buttons, mostly from CW era and post CW, just no photographs available right now.
Don