Hi Bryan,
I can't speak for the Union side but I don't believe there was ever a particular flag for clergy in the southern armies. At one time I thought the Christian flag was inspired by a few of the battle flags of the western armies of the Confederacy. The Missouri Brigade flag and the Polk flag come to mind as well as Breckinridge's Divisional flag all utilize the Christian cross (see the Flags of the Confederacy website). The Christian flag was actually conceived by Charles C. Overton of Coney Island, New York on September 26, 1897. Being a northerner it is not likely he ever was aware of these flags much less use them as inspiration. If by chance you come across a "clergy" or "church" flag of that period let us know.
Cheers!
Mike