Today I dug my first eagle button from either WWI or WWII. Still doing research on it but I wasn't patient enough to hold off on writing the post. The back says JR Gaunt and Son Inc New York. Here it is!
All American made buttons date after 1900. A little background on your Great Seal button; they were first produced in 1901. In 1910 a rim was added. From 1910-1917 buttons were gilt or black, black only from 1917-1923 and gilt thereafter. Your button shows remnants of the gilt so it would date from 1910-1917 or after 1923. The company was in business until 1968.
I just looked at your button again and noticed the "made in England" on it. Even though it was made in England the rest of what I told you remains the same. There are numerous backmarks for the Great Seals and they can make an interesting collection. Good luck in finding more.