That's a nice find. From what I can see in the picture you posted, it looks as if the eagle is holding an anchor in it's talons, thus indicating, to me anyway, a naval origin of some sort. Buttons aren't my area of expertise, so I can't offer up any information on that maker.
It's a U.S. Navy uniform button from the latter half of the 1800's. There is some disagreement as to whether it dates from the Civil War period or is slightly postwar.
IT LOOKS JUST LIKE THE ONE IN ALBERT'S BUTTON BOOK ON PAGE 102 NUMBER 118 BUT THE BACK MARK I HAVE NEVER HEARD OF,IT'S NOT EVEN LISTED IN THE REFERENCE OF BACK MARKS.(BUTTON # NA 118)
This maker/backmark is listed in both McGuinn & Bazelon's Military Button Makers and Dealers, Their Backmarks & Dates, and Tice's Dating Buttons. Nayler's Military Button Manufacturers from the London Directories, 1800-1899 also lists various Starkey backmarks, although none specifically for Samuel Starkey.