There is a Private John Bookout who enlisted from the Russelville area into Company "F" 1st Arkansas (Stirman's)Cavalry Battalion, February 15th, 1862.
Private John Bookout who enlisted in Company "C" 4th Arkansas (US) Cavalry enlisted Nov 27th, 1863 at Dardanelle, Arkansas. Age is given at 45.
So they are quite possibly the same man. However, 45 was above the age of conscription in both armys at the time.