Hm... so wikipedia shows that the resonance form of phenol (a quinone) is part of the mechanism. The amine version of this resonance form (name escaping me) would impart similar stability at the para location, and explain why the reaction tends to be pretty specific for that location.
I'm racking my brain but cannot come up with an example of an azo coupling that doesn't contain a phenol or aniline. It is possible that you just need a group that will stabilize the resonance form enough to make it a viable spot. In that case alkoxy substitution, thiols and even halides may be reactive under the right conditions.