Let's stipulate at the outset that almost everyone on the right you hear talking about the issue of contraception coverage is cynically adopting this position for no other reason than they believe it to be a handy cudgel to bash the Obama administration. (One notable exception is Rick Santorum, who genuinely believes that contraception is wrong, since it unleashes our dirty, dirty thoughts and allows people to have sex without being punished for it. But Santorum is also pro-Crusades, so make of that what you will.) They may be right or wrong about the political wisdom of taking up this fight—a lot depends on whether the administration stands firm and makes sure everyone remembers that what we're talking about is birth control, for goodness' sake, something that outside the ranks of the celibate old men who run the Catholic Church is accepted by just about everyone, Catholics included. But we should keep in mind the principle for which conservatives are now arguing.
Their argument is that a large institution like a hospital or university, if it has a religious affiliation, should be able to pick and choose the laws it follows. In this case, remember, they aren't being asked to use birth control or dispense birth control; all that's required is that the insurance coverage they provide their employees include birth control (free of charge). That's the law, as it was passed in the Affordable Care Act. But the Catholic bishops don't like the law, so they don't want to follow it. According to this principle, religiously affiliated hospitals or universities would be able to ignore any other law as well. Let's say they decided that they didn't like minimum wage laws. They could say that their "conscience" forbids them from paying the janitors and cafeteria workers they employ more than $2 an hour, and it's their prerogative to ignore the minimum wage if they like. After the hospital paints its exterior, it could dump the leftover paint in a nearby river in violation of environmental laws—hey, our scriptures say God gave us dominion over the earth, so too bad.
It's great that the Catholic Church decided to set up a bunch of institutions that perform secular tasks (education, health care) and employ and serve lots of non-Catholics in doing so. But once it makes that decision, it has to abide by the laws of the land. Your local synagogue couldn't start selling energy bars made with strychnine and claim that they don't have to abide by FDA regulations because they're a religious institution. I can't say that my work as a carpenter is affiliated with my one-man religion of Yastrzemskism, and therefore I can steal my neighbor's car to haul some lumber (not that Carl Yastrzemski himself would ever approve of such a thing). Once the Catholic Church moves out of its narrow realm of practicing its religion, it has to be subject to the same laws as anyone else, otherwise every one of us would be able to make our own laws.