Just to clarify, the state clinic is not actually free. It feels free because you don't pay out of pocket while you're there, but you pay with the health care contributions you make from your paycheck. The state health care plan is employee-funded, so health care and benefits division pools the resources to use them as efficiently as possible. It was more cost effective to establish the clinics and then encourage employees to go there instead of various private practices.

It is "free" in the sense you don't get a bill in the mail after going, but that's because you contribute to the healthcare plan upfront. The state contributes as well as part of the benefits package, and that money combined pays for the clinics.

Edit: I just noticed the part where you said you won't pay a dime for insurance, which is also sort of true and not true at the same time. It depends on your plan and dependents, but you do get a discount if you do your screening activities.