A response to "The Leading Edge of Tyranny".
I'd like to say start off by saying that framing this as a question of "free contraceptives" would turn all health care debates into "free X". Health care is not about "free X". It's about paying and contributing to a health care system that supports the needs of its members.
Contraceptives are an absolutely integral part of health care for women, therefore any health care system that provides care for women needs to include them. Any woman age 12 to 50 may be in need of contraceptives, whether for actual conception prevention or managing their menstrual symptoms. Some women need contraceptives to keep from bleeding constantly and developing anemia, while other women need it because they are married* and their finances are not sufficient to support a child.
If you design a health care system that does not support this very important use case for women, or allow companies to remove this from their plan, you are taking away one of the most important features of health care for women under 50.
That's why this has been so irritating to the majority of women. Any woman who works at a company that provides health care but decides their religious beliefs don't support contraception will be forced to pay exorbitant fees to get their insurance elsewhere. It is a big deal for us.
Many people are reasonable and would agree that there are some cases where contraception should be a legal requirement for health care plans where purchase of a health care plan is required, regardless of the moral views of the employer, such as anemia or poverty.
I'd like to take this a bit further.
There are many other reasons than poverty that a married woman may wish to not have children at a particular time. For example, she may believe that she cannot be a good mother because her work requires her to travel all the time. Or, she may believe that she is too old to have children without too high of a risk of having a child with Down Syndrome. Or, she and her husband may believe that they already enough children, and they do not think they can provide sufficient love and attention to their children if they have more children.
At what point would it become the business of the employer to judge whether that reason is sufficient to allow contraception via their company insurance plan? I would argue that any attempt to give the company a voice in this matter adds a horrifying element to the work relationship. In fact, the discussion itself would likely cross into protected territory under federal harassment laws.
Can you imagine discussing with your boss or HR department that you and your wife have a healthy sexual relationship, but she travels too much to take care of children right now? That's preposterous.
If men are unwilling to consider that discussion for their wives, perhaps fathers would be more inclined to take care of their daughters? At what point does it become the company's business that their daughter has need of contraceptives under the company family plan that they pay into?
Admittedly, all this discussion is only relevant given the premise:
"Given that companies must provide health care coverage, must companies provide coverage that is against their stated moral beliefs?"
If we instead decided that that premise was an invalid place to begin because you think that companies shouldn't be providing insurance coverage in the first place, I believe that this issue becomes even more clear.
If we had coverage at the national level that we all paid into, it would be a given that some members may be providing coverage that other members don't believe in. In fact, this is the way of democracy. Sometimes people pay for things they don't believe in because we all have different beliefs. I carry around money that says "One Nation Under God" that was added in the 50s, and I certainly don't believe in that. This doesn't hurt me. Nor does it financially hurt a company to allow contraception in their health care plan for any of the above non-medical reasons. In fact, it would be cheaper on them to allow it because the medical costs of an additional person to take care of are far greater than the costs of contraception. So this debate can not be framed as "put your money where your mouth is", because it is cheaper for the company to allow it. It is about letting the company have control over their employees personal medical care, which is unacceptable to me.
No one is forcing people who don't want transfusions for religious reasons, or who don't believe in using contraception, to use contraception. Instead, we are asking that those of us who do want contraception not be punished for working at a company that believes differently than we do. If I went to work at Logos Bible Software, because I'm a software engineer who can make software that they can use, I expect not to be punished in my health care for doing so, or to be forced to have an awkward conversation with my boss or HR department explaining why contraceptives are medically relevant for me. (I'm not picking on Logos here, they seem awesome, and I have no idea what their health care policies include or don't include. I'm just trying to make this relevant for people who say you should just "work elsewhere".)
Similarly, a Catholic person at a non-Catholic company would not want to be forced to have a discussion with their boss or HR over why they are not making use of provided contraceptives, as their children are becoming so expensive to the company's healthcare policy. I would not want to be a Jehovah's Witness person at a Catholic company explaining to my boss, on my deathbed, why I would die rather than accept a blood transfusion. It's just not their business.
And here's the thing. Catholic women use contraception too.  There is no reasonable argument for allowing clueless male policy makers and exclusively male church leaders  to attempt to dictate conception and family planning policy to women employees, Catholic or otherwise. It is clear to the overwhelming majority of women, Catholic and otherwise, that natural family planning does not work. Contraception and abstinence are the only options that have a hope of working, and YOU try to tell your husband that he can't have sex with you tonight. If you are the husband, maybe you get it? But the point is, abstinence doesn't work either with humans, so...contraception is the only valid method for controlled family planning.
Ultimately it is the woman who has to deal with the fallout of a pregnancy or developing anemia, and women are overwhelmingly arguing that men should stop trying to legislate control of their bodies over to people other than themselves. Money dictates whether women have access to contraception, especially poor women and students, so that's why it is so important that it be included as part of the health care plan where health care plan purchase is required. If I was required to buy into a health care plan for my company, and that health care plan did not include contraception, I would be extremely angry. If I was forced to purchase a health care plan while in college (and I have been), and that health care plan didn't include contraception, I would also be extremely angry. I am in no way dictating that Catholic women should use contraception, but even for Catholic women working at Catholic companies, it should be available to them if they choose, and based on available data, they do largely choose to use contraception.
* Please note that I'm not making a personal judgement on whether or not women should be married to have sex. I just think it is easier to argue with religious people while staying in their moral boundaries. This debate is primarily happening between religious people, so I frame my examples in ways that they have absolutely no reasonable claim that you would be harming yourself by having sex.