I hate giving bad news and do not suggest you give up looking, but I will say that my great grandfather, Dr. John Waller Johnston was also at Memphis Hospital Medical College 1886-1888 and I have not been able to locate any records for him. Some records do exist at the current University of Tennessee College of Medicine in Memphis, but apparently the current institution is not a direct follower of the previous one and not everything passed from one to the other. When I was in the UT library some years ago they did have a few old school Annuals, group photos, etc. but not a complete set. Try
www.utmem.edu or mailing address for the main Admin. office is 800 Madison Ave., Memphis, TN 38103