Mites can hide in anything fluffy used for bedding, not just hay. Hay molds if it's wet, that's why it's not a good idea to use as general bedding where chickens poop. But the nesting boxes should never be wet. If you have wet nesting boxes, you have a bigger problem than the hay, and need to address that. Dry hay in nesting boxes is perfectly fine.
Free option: save your empty feed bags and fold them to fit the bottom of the nest box. Top with nest material of your choice,. If they get badly soiled, simply remove, toss, insert new feed bag.
