There is one coach I've seen mentioned on several sites including ESPN and CBSSports who would be unacceptable to a large segment of the UCLA fan base, even though he is 16-4 in his current position: Spencer Danielson.
Following the team's Fiesta Bowl loss, Danielson stated that his team is "giving Jesus the glory" and advising recruits that "if people don’t like that—don’t come here."
TJJ can add this to his list:
Don't hire a coach who uses his position to promote his religious beliefs.