Legal Question in Military Law in Tennessee

An administrative discharge is not a criminal conviction.

A "bad paper" discharge (resulting in an RE-4 code for reenlistment eligibility) may be done administratively but it may also be the sentence of a court-martial.

There is no such thing as a "neutral" discharge.

If you enlisted (raised your hand, took the oath, reported to basic training) then you were in the military, however brief.

What actual discharge did you receive?

And the question as to whether or not you can purchase a firearm depends on many things other than your military history. Convictions for domestic violence, mental illness, felony convictions (state or military), citizenship issues, orders of protection/restraining orders, can all be disqualifications from purchasing a firearm.