As I have said many times, the only people who should be teaching ESL are native speakers who hold a relevant degree, a teaching qualification such as the PGCE and a classroom based TEFL certification.

If you have a degree in, oh let's say history, something unrelated to the English language, you may as well not have one at all for all the use it provides to the teaching of English as a second language. In that respect I would say the degree holder and the person without a degree are actually equally as unqualified as the other. Simply attending university doesn't qualify you to teach a second language.

I would always take a native speaker over a candidate to whom English was not their first language, simply because thier use will not be as natural as the native speaker.