re: How is this not a violation of EEOC/ADA laws about revealing personal medical information?

I think the other, and equally concerning, question is: Is this a company you really want to work for? They are clearly pandering to fear and intentionally stigmatizing those who don't conform to their social principles.

When it became obvious my previous employer's values and political correctness were no longer in alignment with my own, I left with quickness and found a new career. It turned out to be the best move I have ever made.