They would be relatively old, but doing the math I could see the possibility of a Civil War veteran being at a leadership age in the military.
As a follow-up, there would be media and others alive across these two wars. Was there discussion of the Civil War when America entered the fray for WWI?
There was exactly one. Or two if you use a broader definition. The Civil War generation had aged into retirement by then, but Peterr Hains returned to service.