The degree requirement in many countries, certainly in China and Thailand, is really required to get a proper working visa, not as a teaching qualification. You may have a degree in biology, not very helpful in teaching English. In China certainly, you can teach without a Z, or working visa, and many teachers are told to come over and start teaching with tourist visa's which can be converted to working visa's on arrival, which has not been the case for around two years now. It still happens though, and due to money changing hands local government offices will happily turn a blind eye. Working on a tourist visa leaves you vulnerable to unscrupulous schools though, so should only be considered as a last resort.
Be the change you want to see in the world.