Moving with Pet Bunnies to Costa Rica

According to what I have read all rabbits must travel 'in cabin'
United posts :United allows domesticated cats, dogs, rabbits and birds to travel accompanied in the aircraft cabin on most flights within the U.S. An in-cabin pet may be carried in addition to a carry-on bag and is subject to a $125 service charge each way. There is an additional $125 service charge for each stopover of more than four hours within the U.S. or more than 24 hours outside of the U.S.

In-cabin pet kennels
A pet traveling in cabin must be carried in an approved hard-sided or soft-sided kennel. The kennel must fit completely under the seat in front of the customer and remain there at all times. The maximum dimensions for hard-sided kennels are 17.5 inches long x 12 inches wide x 7.5 inches high (44 cm x 30 cm x 19 cm). The maximum dimensions for soft-sided kennels are 18 inches long x 11 inches wide x 11 inches high (46 cm x 28 cm x 28 cm). There may only be one cat or dog per kennel, and the animal must be able to stand up and turn around comfortably."

You will have to check for International travel.