American thing, they wanted NLRB certification (makes striking legal and a few other things). In the US, the union is also supposed to exclusively represent the entire shop that they organized in. So if the majority didn't want one, the NLRB won't certify, the union won't be able to exclusively represent the entire shop or bargain for workers or strike (legally, sorry meant without scabs you can always strike for better conditions but employers are allowed to replace you without a union).

They could seek some other kind of representation, like IWW organizing or minority unionizing, but there's even fewer legal protections for those people than what scraps of civil rights are afforded to unionizing through NLRB elections.