Ok, so the Boone page on the wiki says that he can bug out and not have any upgraded armor. This is the case with my game. Currently Boone is wearing only a Merc Grunt outfit.

Is it a genuine bug, or is he only supposed to get that armor after his subsequent quests? And if it is a bug, is there any workaround?

Probably a stupid question: have you tried taking it out of his inventory?

He gets naked.

You unlock the armor after doing his second sidequest.

I have only done the quest to get Boone to come along. No companion quests. I have him sporting the new Fall line of Reinforced Leather Armor.

I opened up his inventory and put the armor in there and he put it on. I used the "Let's Trade Equipment" dialog choice. If you just open his inventory form the Companion Wheel like he is a pack mule, then he will just carry it and not wear it.

That's what's supposed to happen. Just put whatever armor or clothing you want him to wear in his inventory. He, like most AI, will always wear what provides the most protection. So you need to make sure that you either take his current apparel or what you want him to wear has more protection.