I'm currently in the process of applying. Submitted everything and just waiting to hear form schools. Some have already provided me with secondary apps.
You will see if the school is particularly picky or not by the secondary app. It will either ask you for the obvious citizenship questions, and if you don't have it: you will stop that application, or it will tell you to provide documents that you may have from INS at the time of an interview.
For the most part, the private schools will be our best bet.