It is not a button... it is a US Army collar-insignia disc. Yours bears the emblem of the Engineer Corps.
Time-dating the version you found, which collectors call a "Type V" disc:
1- Its back has a sheetbrass "plank" insert, with broken off stubs where the two clutch-pin attachments were located. The brass plank was used from slightly after World War 2 to 1970.
2- In the photo, your disc appears to be "slightly" convex, rather than completely flat. If so, that dates it from 1951 to 1970.
The information above is drawn from the following webpage. Go to that webpage and scroll down to "Type V Disks 1942-1970." The first photo in that section shows the back of an Engineer Corps collar-disc, same as yours except the "plank" is iron or zinc (which means that one dates from 1942-1945). Dating Metallic Insignia: Collar Disks