The authoritarians are quite right. There's a lot of intimidation that goes on around the issue of labor organizing. Threats and consequences are appended to how you might vote. No one wants to enter into a voting booth with those sorts of things hanging over you. That's why it's illegal to do any politicking, like handing out leaflets or posting signs or talking to voters or even wearing buttons within 100 feet of a polling place.
And yes, businesses are guilty of politicking too and even threatening pro-union voters with getting fired and losing their livelihood, risking bankruptcy, if they can even qualify, and losing their homes and even their families. So yes secret ballots might help.
But even better would be secret elections. Read my idea after the jump.
The government would gather up all the workers and tell them that neither the union nor the business knows that they are here. You are here to vote on whether you want a union to represent you. At the end of the vote, if a majority vote yes, then they will automatically be in a union. Then the newly created union can begin a process to select an existing union organization to affiliate with. Only then is the company is notified.
Now if they vote no, then nobody is told anything. Indeed the government denies that it even happened. Nothing happens until next year when another secret election is held. Of course the denial part only works for a Democratic regime. Republicans would just never do them.
Okay, so maybe there are holes. But you get the idea.