I'm not finding anything either.
His obituary in the "Atlanta Constitution" indicated he was a Confederate soldier and served four years.
On the other hand, Dr Flowers on the 1910 Census did not indicate if he had been in
the Confederate or Union army, leaving the box blank.
He appears on the 1864 Georgia Militia census, which does not prove service, only that he was potentially
subject to service. It suggests to me Dr. Flowers was not in the service, at least for a time in 1864.
He could have been in a militia unit.