The Religious Freedom Restoration Act, which is at issue in the Hobby Lobby case, stipulates that the government cannot substantially burden the free exercise of religion even in a law that is neutrally applicable (i.e., one not intentionally geared to restrict religious practice).
There is an exception if both of two conditions are met.
First, there has to be a "compelling government interest" in the law.
Second, the law must be the least restrictive way to accomplish this compelling government interest.
It is hard to see how providing free contraception is a compelling government interest in the same way that making sure that we have a vaccinated population is a compelling government interest. Eliminating deadly diseases depends upon having a vaccinated population. Pregnancy -- whether considered a good or a bad -- is not indiscriminately caught via a mechanism similar to that of, say, the measles. Ahem. Not to mention that a society needs children to survive, but it does not similarly need polio to survive.
Further, even if the government did have a compelling interest in providing free contraception (because we all know that we must have sex as much as possible and do so without "repercussions," but I digress), there is a way this can be done that is less restrictive than requiring those who have a long-standing religious objection to contraceptives (e.g., there are some that function as abortifacients) to provide them. The government itself can buy them and provide them.