As a physician Dr. Flowers wasn't required to render military service for either the Confederate government or the State of Georgia. He's not on the 1864 census for reorganization of the Georgia militia, although the name John M. Flowers, age 18 of DeKalb County GA, does appear there.
George N. Flowers may have belonged to the county board of examination, a group of local physicians who issued certificates of exemption for men having physical disabilities that disqualified them from being in the military.