James Thomas Hays and Sons

First two men are Jeff and Joseph, but I don't know what the order is.  Middle man is James Thomas Hays.  Next is Thomas S Hays and last is Benjamin Franklin Hays, my greatgrandfather.

This medal was passed down in our family from the James Thomas Hays pictured in top photo to Benjamin Hays, down to James Thomas Hays Sr, and then to James Thomas Hays, III.  We can find nobody that knows anything about this medal.