Crickets are omnivorous insects that belong to the suborder Ensifera characterized by a cylindrical body and a round head that has longer slender antennae (when compared to grasshoppers) and they often make a chirping sound.
Some of their predators include toads, rats, birds, frogs, snakes, salamanders, lizards, shrews, bats among others.
If you are uncertain on whether rabbits eat crickets or not, the answer is no. They do not chomp them for the same reasons why they do not eat grasshoppers or any other bugs which we have elaborated well. However, let us look at the reasons shortly.
Firstly, you need to know that rabbits are herbivores meaning that they feed on plants and only plants. Their digestive system is adapted to handling plant materials especially those with high fiber and low energy such as hay.
Secondly, they are prey animals that do not have hunting skills to enable them to prey on these insects or any other animal. Therefore, they will not pursue them or any other bugs.
Finally, if you give them these insects, they may munch them. However, they are not nutritionally healthy, and they may present gastrointestinal problems if eaten in large amounts.