Simply stopping work doesn't count, it sounds like you have to go through a formal process of notice, including collective demands, and an agreement to return to work following agreement or a good faith negotiated contract. Of course the benefit of unions is that they have experience in ensuring a strike is legal, and providing legal assistance - in some cases unions can provide financial support for strikers (otherwise you trivially get to a "I just hold out until they lose their homes/children starve" state where strikes are irrelevant)
From various legal firms it _sounds_ like you don't need a union to strike, but your strike still needs to be legal (there are illegal strikes).
Of course it goes without saying, but I am not a lawyer, and am just going from the NLRB site and various law firms websites (on both worker and employer sides).