Have you consulted the book "The Meighen Sept in America and in Ireland" by Bernard Patrick Meighen? Your John may well be in there -- the book chronicles the Meighen family in Pennsylvania and Ohio quite extensively. There is a civil war vet John Meighen in there who would be about the right age -- he was born in 1843 and and enlisted in 1862 -- he lived until 1933! This John's father was named John also, though, not Dennis, though there are a couple of Dennis Meighens in the family, including the brother of the elder John (uncle of the civil war soldier John). Also it appears John lived in Pennsylvania, not Ohio, but there was quite a bit of movement of this family between Ohio and PA (and Illinois and Minnesota, too) so there may be a connection -- worth checking out.
If you don't have this book and want more info please email me at firstname.lastname@example.org and I'll see what I can find --