As far as I know, there is no law against it. Oddly enough, when I was in nursing school, I worked as a nursing assistant. We HAD a union. I never once saw my union rep. It didn't have any effect on my breaks or pay at all. And, it cost me $12 a paycheck regardless of the number of hours at work. My experience with nursing and teacher's unions has been poor. Of course it could be that I lived in a right-to-work state at the time. Unions like the United Autoworker's Union have always seemed more effective to me.
That said, my good friend Linda organized one for the audio visual and teacher's aids in her school district. She started by gathering together a group of like minded co-workers and bringing their common grievances to management as a group. The union grew out of that effort. So, I would do the same: gather together your coworkers, list your grievances, and set up a meeting with management where you can express your concerns.
If you approach management with something like, "We'd like to make this a better place to work for everybody and put patient concern above all. Here are some things that we think will attain those goals........"