Nice bit of research on the militiamen posse. You are correct the the 28th EMM is primarily an Osage County unit, while the 42nd EMM is a Cole and Miller County unit.
I looked at the online military service records for the Bax men in the 28th. They were all in Company F, whereas Frank Bax enrolled in that company at the Osage County seat of Linn in August 1862 at the onset of the EMM program, while Henry and Joseph enrolled later at a town whose name is hard to read or understand. It looks like "Kozltgtown" or some such. I wonder if that place is over west toward the Miller and Cole County side of Osage County. If so, perhaps Ben Bax is actually Henry or Joseph Bax and "Ben" was either the middle name of one of them or the nickname.
Thanks for helping on this.