I believe these questions or similar questions can be asked post hire in USA as long as they justify it for medical purposes, and are asked of all potential hires. And, the final result can be them backtracking on the offer if it was contingent on the exam. They have many ways of quietly discriminating against those that honestly acknowledge a mental health history of any sort. A biggy is for them to request an outside doctor to certify you as fit for he job.

You might want to get clarity from an agency or lawyer before you make a decision.