In Australia we keep pet crayfish but we call them "yabbies".
Yabbies are mostly vegetarian and will eat thinly cut vegetable such as celery, potato, pumpkin, apples and other fruits.
I have read that it's recommended that you feed them a regular supplementation of chicken pellets or a little lean meat.
You can also buy yabbie food. Here is a link:
Hikari Crab CuisineThe crayfish pellets would probably be used instead of the meat and chook pellets