Some crickets have real big mandibles and I'm pretty sure they could hurt your animals if not careful. Field crickets for example (the black ones) those things can bite pretty hard if they want, especially the big ones. They might be alright for some bigger lizards/chameleons but not sure if I'd risk it with a frog, maybe a pyxie but why chance it? Try something like hornworms or silkworms if you're looking to add something to their diet.
I've never heard of or seen a Jamaican giant cricket, so I have no idea how big their mandibles are. I'd just assume from the name they might be big..
