I am using Wikipedia's 1777 and 1778 lists of events to try and narrow down where he might have served, and what he might have been doing. I can't find any military activity around Fort Pitt in what is modern day Pittsberg during those years though the American Revolutionary war was being fought by Washington on the other side of the state. In 1774, there was a massacre committed against the Mingo/Minge people that lead to "Dunmore's War of 1774". On FamilySearch, there is an unsourced note that says he died of Cholera: "RECORDS OF DON THOMAS !Ft. Pitt is now Pittsburg died of Cholera". Also, an unsourced note says, "Age suggests Captain John McCulloch is the father of those children herein listed, but this is not absolutely established. His brother, George McCulloch, could have been the father. Appendix 017" Which seems to be an excerpt from an uncited book.
There's also this: http://lotsofsites.com/gen/mcbible.shtml
His parents appear to have been married in Ireland (1724) and probably lived in Gloucester, New Jersey during 1726 when he is recorded as being born which would be adjacent to the Pennsylvania Campaign of the American Revolutionary War in 1777-1778.
Thank you for the records. Hopefully they'll be helpful in resolving this.