Found this today in western NH, in a corn field. Appears to be a domed button. There is an eagle on it with stars to the top. It’s a bit smaller around than a dime, and is about 1/4 inch tall.
That your button has an iron back usually means it's not military. However, on some of the hats, the button could have an iron back as it would only rust on the leather band. Your button would be an officers button if it was for the hat.