The Catholic Church teaches that artificial contraception is intrinsically wrong, and that its wrongness is a matter of ordinary morality--"natural law"--which in principle a reasonable person ought to see, like the wrongness of theft, adultery, or murder.
So, although the HHS mandate does indeed violate religious freedom, more basically it is an attack on conscience. It is the state compelling citizens to endorse and cooperate with something which is intrinsically wrong.
Analytically, then, the mandate is similar to a commanding officer who tells a soldier to commit an unjust action on the battlefield. If the soldier were to refuse to obey, as we all know he should, his defense would properly appeal in the first instance, not to his religious liberty, but rather to his conscience.
It is important for Catholics to appreciate that from an historical perspective the Catholic Church's teaching is not the anomaly. The common, universal view of all responsible institutions in the West, until 1930, was exactly that contraception is wrong as a matter of ordinary morality. In that year, however, the Lambeth Conference of Anglican Bishops created an uproar by tentatively suggesting that some married couples might, in some limited circumstances, use contraception without doing wrong.
To see how anomalous the Lambeth view was in the context, one only need turn to the pages of the Washington Post, which in an editorial and article in 1931 gave its views on a follow up report by a committee of the Federal Council of Churches, which had reached a conclusion similar to Lambeth's.
"The Committee's report," the Post editorialized, "if carried into effect would sound the death-knell of marriage as a holy institution, by establishing degrading practices which would encourage indiscriminate immorality." (Did Pope Paul VI in his encyclical, "Humanae Vitae," write anything stronger than this?) Also, an article in the Post quoted a Dr. Walter A. Maier, of the Concordia Theological Seminary in St. Louis, a Lutheran institution, who said that contraception "is one of the most repugnant of modern aberrations, representing a twentieth century renewal of pagan bankruptcy."
Dr. Maier's well-informed opinion was simply the common view among Americans at the time. The Catholic Church is singular in being the only major institution which since then has preserved its moral sense.
The natural law never goes away. It niggles away in the back of one's mind or even at the edges of a society which has rejected it. I cannot tell you how many times after giving a talk on contraception some woman late in years has approached me and whispered in confidence, "I never felt comfortable, I always knew there was something wrong, when my husband and I decided to stop having children." Similarly, the uproar over the HHS mandate goes far beyond any reaction that could be predicted to an infringement, say, upon a religious dietary law -- which confirms indirectly that we are indeed dealing with a matter of "natural law."
So the first thing that Catholics need to appreciate is that we are living in topsy-turvy times, when the Catholic Church has remained consistent, while many other major institutions of our society--as they have as regards abortion and marriage--have become unbalanced and even inhuman. Catholics ought to love their Church more for this, and practice their faith with greater fervor and unswerving devotion. Dissenters ought to be recognized for the quislings that they are.