Can a boss have the right to keep adjusting a worker's pay?

Yes. The boss can do that. If your wife doesn't like it she can work somewhere else.

It's only illegal if the boss signed a contract with a union saying he can't adjust the pay.

Many different deals are legal. You can agree to do work for a fix amount, if you wish. Did she know that it was going to be like this when she entered? And, maybe more importantly, are there laws that describe how workers should be paid after so many hours of work or did she signed a contract that specifies it?

In the USA, without a union contract, an employer can change your job description, cut your pay, eliminate benefits, or even fire you on a minutes notice and there is nothing you can do about it unless you can prove it was done because of her belonging to one of the protected classes.

If she doesn't like working there, her only recourse is to find another job.

if she doesn't like the pay, then leave. You have to tell them," I'm leaving unless you pay me more".