I know this is an old thread, but it is important to know that trying to get any animals on to the island puts you into a boat load of red tape. Cats and Dogs go through a quarantine process that takes 180 days where they will be isolated at the AG facility, but can be taken care of before you try to come over. As for the other animals, it will be a case by case decision from the Hawaiian Department of Agriculture.
THINK and PLAN well ahead if you want to get your animals onto the Islands of Hawaii. Any quarantine costs will be borne by the owners, and non approved livestock, animals or plants will be refused or destroyed and necessary permits must be obtained prior to coming to the Islands
This is a good place to start your search: http://hdoa.hawaii.gov/ai/aqs/animal-quarantine-information-page/
They are currently refusing all fowl except chickens without an import permit and very few of them are allowed since a bird flu outbreak in 2006.
You can find a lot of animals on the island already like peacocks, guinea fowl, some breeds of turkeys, all breeds of chickens (they are still allowed to be shipped in under tight regulations), cattle & horses are limited in breeds here, not sure why yet. The biggest concern on the islands is when the pets/livestock get loose and become feral. We have wild cats, dogs, pigs, cattle, donkeys, and turkeys all wild and all a problem in some way in parts of the Islands.
There is a list of prohibited animals that are not allowed for import that you can find here: http://hdoa.hawaii.gov/pi/pq/import-program/animal-guidelines/ and in case, a list of Non Domesticated Animals and other organisms as well here: http://hdoa.hawaii.gov/pi/pq/import-program/pq-non-domestic-animal-and-microorganism-lists/
The moral here is do it right and the feeling of Paradise can be seamlessly continued. Do it wrong or not in time and your pets may be left behind and sour your move from day one.
I just moved to Hawaii island last month with my 3 dogs... it took me six months before moving to complete all the requirements. It cost almost $4500 to get three dogs here, (two which were over 75 pounds with their kennel which increases the price greatly). The blood tests alone were $600. It’s not a small deal at all. I chose to find homes for my chickens on the mainland and start over in Hawaii once I realized how much just my dogs would cost to move.