I have been continuing my research and while still not proving a link between the two Menz mentioned I have been able to obtain further information on Abel's birth.
He was born Johann Abel MENZ on June 21st 1835 in Albrechts near Suhl. The parish that contains his records is Suhl-Heinrichs in Suhl.
I got this information after sending out a request on a mailing list for Thuringen. Someone contacted the parish and the parson looked up the records on the spot. I am very grateful to him, as now I know exactly where to look for the connections I am trying to prove.
I also know that Abel was the first child of Johann's second marriage.