First thing to ask is what state you live/work in? Is it a right-to-work state? If so, then they can fire you or choose to not promote you for no (reported) reason at all, which very likely means you have no legal recourse. If they were to come out and directly say in documented way that they will fire or not promote you if you don't use this app, that might be different. You'd need to talk to a lawyer who is familiar with laws in your state. But you'd also need documented evidence of this, which means emails sent stating this, or a recording (keep in mind if your state has 2-party consent laws) of a higher-up saying it.
If you're in one of the 27 Right to Work States, though, there's likely very little you can do about it short of finding a different job.