I live in the US Virgin Islands and travel often back and forth from the mainland with my two small dogs. All your dog would need is a health certificate dated a maximum of 10 days prior to travel. Make sure the vet does a domestic health certificate since we are part of the US and NOT considered international. Your only problem may be on the outbound when you return home as you will need a health certificate dated within 10 days of travel. So if you only stay a week and get the certificate in the states a couple of days before travel, you should be fine. The certificate basically is proof of an up-to-date rabies vaccine (since we don't have rabies on island). They rarely check them, but you won't be able to travel if you don't have one.
You may also run into trouble with taxi's transporting you from the airport to either Charlotte Amalie harbor or Red Hook where you will catch the boat. Some taxi drivers may try to charge extra or be disagreeable about transporting a dog (no matter how small). So when getting a taxi, keep the dog in its carrier and avoid mentioning it at all. Hopefully your dog won't bark and the driver will never know.
Have fun in beautiful St John!