As Bryanhashemi indicated, very-precise measurement of the buttons' diameter (in millimeters) is needed for certainty in identification. The Albert button-book lists only three sizes (28mm, 23mm, and 16mm) for GI-105 (1902-1910) and GI-102 (1910-to-present) Great Seal buttons which were actually issued by the US Army.
Also, Mr. Albert's book mentions the existence of ironback Confederate buttons specifically, by listing various ones as a tinned-iron-back (t.i.b.) button. But he does not mention any iron/steelback Great Seal buttons.
I suspect precise diameter measurements will give us a clue as to whether or not any of these three buttons are military-issue ones, or is a "Fashion" button.