Everyone has the right to organize or not...it should not be mandatory to belong to a union in order to work in America..IMHO.
You are right, I oversimplified. The actual implementation would be more flexible - you could ask for conscientious objector status for such case like:
Being a Objectivist or Libertarian or Anarchist or Free Market fundamentalist.
Being a Linux consultant who does not take bath
Your religion does not permit you to belong, like Cult of Management Worshipers for example.)
Having a case of ochlophobia (fear of crowd)
I am sure you would find some category to save you from this calamity.
Of you course you would have to forgo some privileges the union members would enjoy, but it will be small price to pay in you eyes.