Yours is the style used after 1902 and is referred to as the "Great Seal". You see this design everywhere. It is used as button design hat pin, front of the presidents podium, etc.
I've heard of people using something called aluminum jelly to clean civil war buttons. I've seen some before and after pictures. It does seem to work well. Although I've never tried it myself. Like was said before with common eagle buttons just use soap and water. They may be common but I sure love finding them.