The main problem for many of these birds is disease (types of avian pox and avian malaria) they have no immunity to. These came when mosquitos of the genus Culex were introduced to Hawaii. These mosquitoes are very widespread around the world but weren't naturally in the very remote Hawaii islands. They are the carriers for many more types than the types of avian pox and avian malaria that have caused havoc in Hawiian birds. Bringing rare native Hawaiian birds to places like mainland USA or Europe where there are even more diseases they are unlikely to have any natural immunity to could easily cause even more problems for them. If you then planned on bringing them back to Hawaii (because you had some success with captive breeding or failed and wanted to save what could be saved) you would risk having some of them being carriers of new diseases and then introducing yet another problem to the Hawaiian wild birds. Even if the species had immunity to the new disease it could still introduce it to some of the other Hawaiian species that don't. Just like some Hawaiian honeycreepers now have immunity to avian malaria but others don't.
Unless someone could come up with a very very good reason to bring some to the mainland and could present good arguments for why they would be able to do things there they can't do in Hawaii you can also be totally sure US Fish and Wildlife Service would not allow any of the rarer species of Hawaiian honeycreepers or thrushes to be exported. Many are listed on the Endangered Species Act.
Species listed in Hawaii based on published population data