Only if you own a car and drive it on public roads.
For all intents and purposes, auto insurance is effectively mandatory for most Americans. It's for a similar reason that many states revoke driving privileges for parents who don't comply with child support.
Texas, for instance, doesn't actually make you buy insurance. You only have to prove that you can be financially responsible for the damage you could cause [1].
Why is this relevant? You made the claim that "The sole purpose of the mandate is to make free-riding on the insurance market unlawful"
The medically uninsured only pose a financial threat to themselves. It's only through laws that make taxpayers responsible for the medical costs of the uninsured that they pose a threat to the rest of us. This mandate won't change anything in that respect. It will just make the line from taxpayer money to person who can't pay for his medical bills harder to follow.
[1] http://www.dmv.org/tx-texas/car-insurance.php
edited for clarity
Mandatory coverage is orthogonal to the question of whether health insurance is a proper insurance scheme, or some kind of social safety net entitlement, or a utility. The sole purpose of the mandate is to make free-riding on the insurance market unlawful; it's a mitigation for an asymmetric information problem.