The policy isn't specifically illegal as far as I know.

It can potentially be illegal due to the subjective nature of approving and denying request. For example if someone gets approval for a religious event, but you get denied for a secular event. Any favoritism towards a protected class makes this policy illegal. Proving it in court is difficult unless you have a smoking gun.

If it were me my request would always be "for a religious event." No I'm not giving more details. I need time off for a religious event. Deny or approved?

But you're not me, so it may be the hill to sacrifice your income stability on.

Illegal? Probably not. Asshole move? Most definitely. I think the reason that they're cracking down on this is to avoid potential overtime payments but it doesn't make it right or fair. You don't owe an explanation to anyone if you've been responsible enough to cover your shift. That much said, the only way to be sure is if you file a complaint with Colorado's Department of Labor and have one of their attorneys or Labor Law Enforcement Officers go over your claim. It could very well be considered an unacceptable breach of privacy and this restaurant could face punitive fines for behaving like this but I am neither a lawyer nor a resident of Colorado.

Never ever ever ever ever give your bosses more information than they need. "I have a personal appointment that cannot be rescheduled" "I'm not comfortable discussing that"

I don’t think it’s illegal but several things are true

1. It’s delusional to think it’s ok to treat grown-ass adults like children in this way
2. That type of delusion is 100% corporate culture in retail and food service management. I was a retail manager and it’s scary how skillfully they normalize abhorrent behavior because we need team players and other such gaslighting.
3. Even given item 2, it is clearly a very bad idea for a manager to subjectively approve or deny shift switches based on the reason for them. Soooooo many HR pitfalls that could lead to accusations, lawsuits, fines, etc.

Is it illegal to ask why you need to switch? Probably not, though it’s a bad idea (for them). Can they legally require you to tell them? I don’t think so. Can they fire you if you don’t follow the policy? Depends on the state laws, employee handbook, and lots of details.

I’m not a lawyer and I’m not your friend but these people are clowns…they are trying to run their store the (wrong) way they’ve been taught. The policy is silly and wrong. You don’t have to tell them shit. Tell them it’s none of their business or tell them you are sacrificing a baby on an altar made of cat hair, or you turn into a werewolf on Thursdays. Be nice to the clowns (they may be sad inside) but don’t respect them if they behave like idiots.*

*you may get fired