It's the same place under State & Local > Other > MA Schedule HC.  So long as the insurance your client had met the MCC, why should it matter whether it's private?

Ah, I think what you're looking for is the provider info.

If that is the case, on the same input screen for Sch HC, you'd find a blue hyperlink in the top right corner in tiny prints for Provider.  Click on that and it'll bring to to the input screen for insurance provider.

That's a typical setup in ProConnect Tax.  Very often, additional details are to be entered using the link in the top right corner.