Can military personnel land their own private aircraft at military bases?
As someone already said, 32 CFR 855, Table 1 states:


  Service members, operating their own aircraft, leased aircraft, or other available aircraft for official duty travel (temporary duty, permanent change of station, etc.) or for private, non revenue flights


This of course means you still need to contact base ops for a Prior Permission Required (PPR) of some sort (if required).  If the base has an Aero club, then your logisitcal considerations are much less (most aircraft probably don't want to refuel on JP-8).  The list of Aero Clubs seems to have decreased since I was in though.

The Flight Infomration Handbook will have all the base requirements listed in it.