How would the government require you to actually use your medical insurance? You can always pay out of pocket, if you prefer, under any interpretation of the current law, couldn't you? The government only requires you to be eligible to use insurance - but that would be true even if it financed the national health insurance out of the general budget.
Insurance is about taking what would be a payment for a service, and turning it into a payment for managing risk. So, in that sense by owning it, it's used because it's acting as an insurance for potential needs. The government or another third party paying for a service isn't really insurance in the same sense.
Really, what it comes down to is people have a certain amount of money that they can put toward their health or the health of those they care for, and the mandate requires that this money be put toward a qualified insurance program rather than other arrangements. Not using an insurance that they have bought would only mean they are giving their money to a corporation that they may not want to support, and would have to come up with the money from elsewhere.