Since my last post I have done additional research mainly at the NLRB website (nlrb.gov). The first thing I found is that "Regardless of immigration status, you have the same rights under the National Labor Relations Act (NLRA) as all covered employees". The next thing I found is that it is illegal to use an employee's immigration status against the employee. So illegal employees cannot be exploited, but the US can and is!
So, obviously, my previous post was incorrect. MY APOLOGIES.