Well, yes they can and will snag them. If the nail breaks off down below the skin line, then it will not grow back. My cham has one nail that has been missing since he was a baby.
That is the big dilemma when picking a cage. Fine mesh (aluminum) will remove all of the toenail for the most part if climbed on regularly. While hardware clothe wares the nail down quite a bit however does not completely remove the nail. There is a new kind of durable fine plastic screen that seems to not harm nails. However feeder insects could possibly chew and ingest it before becoming a meal for a chameleon. Meaning a crick gut loaded with plastic is absolutely not an ideal food for any chameleon. I would watch for screen climbing and got from there.
Bird cages won't harm the nails. All of mine are in wire birdcages. You have to cup feed though because your feeders will escape. Also, my ambilobe has learned how to lift the door to escape, so in some cases you may need to install a latch.