I think you'll find most reputable airlines have embargoes on certain types of animal. CV for example (as far as I can remember) won't fly frogs (legs...yum yum) or monkeys (testing)...
Flew a whale from BCN to SAN (from their aquarium to Sea World's aquarium, a captive born white rhino from DUS to JNB (to be let free in Namibia). Pregnant (milk) cows, day old chickens, horses are all "regular" cargo these days.
I think the argument would be similar to a pilot refusing to fly a certain type of pax or saying "I'm not flying to XXX because the govt is repressive".
Do you have the "right"? Sure you do. But then the company has the right to say "Thanks, you can leave now". During GW1 an L/M I knew was let go for refusing to fly "military equipment" as it suddenly turned out he was a pacifist. "But Bob (name invented), these are only gas masks to be distributed to civilians". He wouldn't have it, no way. He was invited to leave.