First... nope, your brass 1-piece flatbuttons with an eagle in the backmark were not made as Military buttons. That being said, we know that a few plain-front ones did get used on Local Militia uniforms when the members were not rich enough to afford State Seal or City Seal buttons.
Theoretically, the eagle was included in a backmark by American button-makers to show their product was US made when many Americans were boycotting British-made products for about a decade after the end of the US-versus-Britain War Of 1812-15. But we also know some British button-makers falsely included an American eagle in their backmark to avoid the boycott. You've got an example of that... your brass flatbutton with an American eagle and the word color spelled the British way, as colour.
By the way, your button whose backmark says "Treble Plated" has what is called a Basketweave pattern on its front, which is a common Civilian design.
A little more info for you about your buttons:
In brass 1-piece flatbutton backmarks, the word "plate" always meant silver plated, and "gilt" always meant gold plated.
