Do companions' weapons degrade in Fallout: New Vegas?

This was changed several patches ago and I never updated my answer.

As of a patch some time ago, companions' weapons degrade per attack at the same rate as the Courier's.

Armor degrades based on damage taken.

As I said in the post the comment was made on, I suspected that this was possible, but couldn't prove it. However, I think I found evidence that it happens just now, at least on Hardcore mode/Very Hard difficulty.

I gave my companion a Super Sledge and got into a fight with several Legionaries, who typically carry machetes and magnums. I didn't see any with Super Sledges, though it was nighttime in-game and I could have just missed it. Sometime during the battle, I heard my companion yell "I'm gonna need a new weapon!" When the fight was over, I went back to the spot where my companion had been. She was back to carrying her default weapon, and I found a Super Sledge on the ground with an empty CND gauge. Oddly, its hover text was red, but I didn't suffer a karma penalty for picking it up.

I was able to put the broken weapon back in her inventory, proving my assumption in the previous post wrong. She wasn't able to equip it, but I was able to take it back out of her inventory later to transfer it to a container.

I gave chainsaw to Veronica and when we fought some ghouls she broke it.

Yes. I gave the Plasma Caster to Cass, and she rendered it useless within a few hours of gameplay. Costed a bomb to repair it.