All companions will occasionally pick up weapons (particularly during combat) if they're in reach and the weapon does more dmg/dps than what they're using. Some people seem to experience it a lot more than others, tho...it's only happened rarely for me. I suspect a lot of the time you/they just aren't in the right circumstances for it to occur.
One time at that camper spot near Searchlight, where you get a small ambush, I was attacked by the usual bandits that show up PLUS a squad of PO'd Legion. Boone was hiding/sniping from inside the camper...there was a tiny break in the action and he ran out of the camper, stood over one of the Legion corpses, stooped & picked up a 10mm machine gun and starting using it. The funny thing was that the corpse only had a tiny bit of 10mm ammo (I guess) so Boone fired it a couple times then hollered "EMPTY" and switched back to his default rifle.
And yeah, companions love the All-American.