Are my managers right or wrong?

It depends on how you look at it. It is initially their problem - if you want to call it that, two choose two people to work. If you are told you are one of those two people (and not given the day off), then that is your problem. Your problem because you have a responsibility to show up to work. If you leave them out in the cold, expect to be fired. An employers is NEVER required to give you time off/approve a time off request. In fact, even an annual vacation - they can insist you take it at a certain time. That's life when you work for someone else. They make the rules. You follow them.

If, the 3 of you cannot decide who works that day, and if, the manager was to decide between the 3 of you, my guess it will be YOU working.

Telling management that supervisors and other managers to work, just because you asked for a day off, would really P me off.

The store must be run with enough employees.

There is no labor law that says an employer must give you a day off, if you request it, even if you asked a year in advance.

Their problem IS your problem since it is your job to do as management tells you.. sorry but if you want a paycheck that's how life goes. I missed many vacations and holidays with family when people got sick or management wanted more people working. That's life. You want to be free of these problems you can always be a homeless person. Otherwise most jobs will have similar situations pop up and you just have to work.

It's your problem. They are correct, a request for time off is not a guarantee.

Your problem. Management can deny your time off request for any or no reason.

The nature of retail is that your management sets the schedule at their whim. Whomever's "problem" it is, they are still the managers. Maybe they aren't good managers, but they are the managers.

Hmm I think its unfair, you told them before and they reprimanded their judgment. It also looks like that place you work has no values for the little people who run the show my guess is to work but if your undervalued, apply elsewhere for a job.

This is why we have unions.