Legally, can an employer ask for information about a medical strategy that is not used on company time?

I don't believe your Dr. can answer their questions without your permission and they are not going to credit what they can dig up on the web because they want information that pertains to you alone. You need to consult a lawyer before going forward and no matter what you do they will be looking for a way to get rid of you because you are a potential source of controversy and head aches for them. Sorry, not legal or fair but employers are never worried about that when it comes to employees rights.