Best to avoid piggybacking on someone else's old thread.
Note employer is *asking* for this info, and yes, it's free to demand it. Doesn't mean you have to cough it up. Obviously, it's up to your wife to decide whether she wants to cough up that info; if she's okay with it, that's all you really need worry about it (HIPAA applies to the medical provider and the employer if the employer is running it's own health insurance plan/is self-insured; meaning, if self-insured, the employer isn't entitled to broadcast private medical details it gleans from medical claims/records to every Tom, ***, Harry).
If employer is subject to FMLA and you eligible, and your wife's serious health condition lasted beyond just the two days in the hospital in terms of medical treatement (not necessarily in-patient) and a doctor is willing to do a medical certification that your presence was necessary for whatever reasons, then you can suggest to the employer that it's free to ask for a medical certification. It doesn't have to go into specifics about the illness, etc. to comply with FMLA regs; the average doctor will be familiar with the fact that the patient will need to okay what info goes on it.