There is nothing that directly speaks to soldiers carrying a privately owned firearm in either the UCMJ or AR 670-1 while in uniform that I could find. Certainly, it's strongly not recommended. Change into civilian clothes after drill and gear up.
The only thing I could find is this:
Carrying of Firearms and Use of Force for Law Enforcement and Security Details.
See the 5th "Bullet Point" Please note this was meant for Military Police.Chapter 2 - 2 talks about "authority" on who can carry.
o Prohibits the carrying of non-Government owned or issued weapons or ammunition (para 2-6).
Basically it boils down to several things. When you are in military uniform you are on the clock. You are no longer "Citizen Joe or Jane." You are under the control of your superiors. UCMJ action applies to your actions from this point.
1) In uniform you are not "Authorized" to carry - see Chapter 2 - 2.
2) Even if you are military police, you are not authorized to carry a "non-Government owned or issed weapon or ammunition."
Yes, most Reserve and Guard locations are not on a military base so it's not so clear cut.
If you do enter and pass through a gated entrance to a military installation then those rules apply.
IMHO: Don't carry it while in uniform.
And don't go asking for permission. This is clearly a don't ask as you surely don't want to know the answer from your superiors. :?