In a union only audition the last thing they want is a non-union person to walk in and waste their time. Not saying you're untalented or unworthy of being seen, but if it's a union only audition it's because they can only cast union people in the remaining roles or they plan on non-union auditions at a different time.
Equity Membership Candidates can sign up for Equity auditions usually, but sometimes even if you have a spot they end up saying no EMCs that day. This is the rule bwaylvsong is referring to, and is for EMCs, not straight up non-union. Sometimes you might get lucky if the unions go lax and you can slip in as a non-union non-EMC, but I wouldn't test that luck if it's going to be your first impression with the theatre. You might get lucky, or you might make a bad impression. Who knows.
Not being harsh, just being realistic with the business side of the art. Your best bet is usually calling the specific theatre for audition information.
"I am ready to disclaim my opinion, even of yesterday, even of 10 minutes ago, because all opinions are relative. One lives in a field of influences, one is influenced by everyone one meets, everything is an exchange of influences, all opinions are derivative. Once you deal a new deck of cards, you've got a new deck of cards."
— Peter Brook