One of the reasons pro-lifers - regardless of their church membership - are concerned about the Health and Human Services ruling is because it's not only birth control that is of concern, but drugs that actually kill embryonic humans. The HHS issues a ruling that compels religious organization employers to provide health insurance that includes birth control. Even today's modification of the ruling still amounts to requiring religious organizations to tell employees where they can get birth control that can, and in some cases does, cause a embryonic human to be aborted.
There is serious concern that birth control pills may cause abortions and that some "birth control" works after fertilization. Not only Catholics are concerned, but all pro-life religious organizations.
The evidence is unclear whether birth control pills cause abortions as a back-up mechanism if fertilization occurs. The Life Training Institute has a helpful summary of the evidence here. Since it's plausible birth control pills function as an abortifacient, pro-lifers wish to err on the side of caution. That's the issue of conscience at stake - not providing the means to possibly kill embryonic humans.
Randy Alcorn has researched this extensively and concludes there is evidence that birth control pills cause abortions. You can read his case here. Most people don't realize that the possibility exists.
So the issue of conscience here isn't just a Catholic concern over birth control, but a pro-life concern over providing the means of killing a human being that has already been conceived.