It is a screw back lapel button. The screw nut was marked "WHITEHEAD & HOAG CO, NEWARK, N.J."
Some history:
During the Second World War Whitehead & Hoag employed over 400 workers but it could not successfully grow from a mid-sized family-owned company to a large corporation. In 1953 the last founders' family member with an active interest in the company died and the firm started deteriorating. One or two profitable years would be followed by several years of financial losses. In 1959, a long-time competitor, Bastian Bros. of Rochester, N.Y., bought the business.
It appears that more than a little vengefulness were at work in winding down operations. Rather than merging Whitehead & Hoag into Bastian's business, the new owner ordered the factory to be sold and all dies and records destroyed. Thus the carefully archived records and dies of many beautiful medals were lost.
The Armory is now the Springfield Armory National Historic Site, and is managed and operated by the National Parks Service. You might have some luck in contacting the NPS.
Don in SoCal