In your natural-lighting photo, the "Great Seal" US Army button on the right looks like it might have what is called "black finish" on it... but doesn't look that way in your incandescent-lighting photo. If it does have "black finish," it is from World War One to 1923. If it doesn't, it is from 1923 through 1942... because in 1943 the Waterbury Button Company changed its name to Waterbury Companies (so their backmark became Waterbury Co's).
Your other "Great Seal" US Army button can date anytime from 1923 to the present. I can't give you a narrower time-range without knowing exactly what its backmark says. Please do the best you can to read it, and tell us what is.