I have seen a lot of threads with this going on and i thought i would just post up what i do in this situation. When i give items to boone i notice that sometimes he looses his rifle. Here is what i have done to fix this every time it has happened to me.

Possible Fix: Talk to Boone and while in the setup screen click on the Melee/Ranged button, by this i mean Turn him to melee then switch him back to ranged and for me at least it comes back everytime.

I hope this helps all of you and doesnt get pushed to where it wont be read if it does.

Boone never loses his rifle nor run out of ammo, but if you give him a new weapon he will use that instead.
However early to mid there are no guns better than boones gun, so just remove any weapons you have given him and he will use his sniper rifle instead.

Well a lot of people have been having a problem with it vanishing and this is what i have been doing to bring it back.