And the answer is......maybe.  Sorry, black and whiters, that's just the way it is.

It depends on the type of free or in this case, discounted care you receive.  So as an example if your employer has a clinic where you can go for a flu shot, or say get your allergy shots, or maybe to get an aspirin when you have a headache, or maybe to have a wart looked at or maybe get some antiseptic and a bandaid if you cut yourself opening a box or something you would still be eligible for an HSA.

On the other hand if you work at say, a hospital, and you get your medical care for free or at a deep discount, sorry Charlie (or Charlene) No HSA For You!! 

Sorry, couldn't resist.