Two caveats before we go any further: first, I'm assuming that Hobby Lobby and the other companies involved are sincere in their faith. If they aren't, then it's just another political lie, any one of which is as bad as another when it comes to shaping policy. Secondly, I am against Hobby Lobby etc. in this case - I agree with the author of that article that there is no biblical justification for opposing abortion, and even if there were, I don't think these companies should get an exception from the law.
(Also, the specific defence offered by the appeals court is ridiculous - basically, they appear to be treating a corporation and its business practices as a 'form of evangelism', which is horseshit).
I do want to try to make a slightly more neutral analysis of what's actually happening here. The reason that, to an atheist, the idea of a religious exception to a law is so abhorrent is that it seems to contravene the First Amendment to the US Constitution, because it is a law 'respecting an establishment of religion'.
There are two points to raise here. The first, obviously, is that the constitution is subject to amendment and reinterpretation in all sorts of ways, and should fit the will of the people. If Hobby Lobby etc. feel that the constitution should be changed, they have a right to challenge it. That said, I think there are very good reasons for the First Amendment, or something like it (I come, after all, from a country where Bishops still sit in the legislature).
But here's what the First Amendment actually says:
"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances."And what Hobby Lobby could claim, I think, is that the law requiring them to provide funding for abortions is prohibiting the free exercise of their religious beliefs. They believe (again, assuming they're sincere) that abortion is the murder of children, and thus prohibited by their religion - and, indeed, by any sane morality ever. They believe it's evil.
And you can say, as the author of that article does, 'well, that's just what they believe, and people can believe anything without that making it right', but this completely misses the point. All judgements about what is and isn't evil are belief judgements. If there are facts about right and wrong, we have yet to discover or agree on any of them.
(This isn't to reject the science - there are facts about what contraceptives are and are not abortifacients, and nothing Hobby Lobby's owners believe can change that.)
Let me construct an analogous case with a politically opposite situation (and I apologise for how hazy my knowledge of actual US law is in this instance). Say that at some point in the future, the US goes to war. You oppose the war personally, but accept that (unlike, say, the Iraq invasion) appropriate democratic process has been followed. Then the government institutes a draft and you (or your child, or friend, or whatever) is called up. Do you allow the government to force you into acts of violence you believe to be immoral?
Of course not. Or at least, it would be an act of severe moral cowardice to go along with it. You look for due-process ways of opposing the government's decision. In the most extreme case, you can renounce your citizenship, but nobody expects you to do so as your first course of action.
And as far as I'm aware, Hobby Lobby are following due process. In the first place, this is a debate about the interpretation of the 'prohibiting the free exercise' clause of the First Amendment. The Supreme Court in the US is, at least by tradition, responsible for the interpretation of the constitution. To get access to the Supreme Court, you have to go up through all the lesser courts. To my knowledge, that's what Hobby Lobby and the rest of their group have done.
Again, I'm not saying that their case should succeed (unless it is, at the widest level, the will of the people of the US, but if it is then the people of the US need to think again about why the First Amendment is so important). But the kind of knee-jerk reaction in the article I linked at the start is not helping and is arguing in the wrong place.
The argument I would be making against Hobby Lobby is that whatever the personal beliefs of its owners, their corporation is in the public sphere (i.e. affects many people besides the owners and thus exemptions for personal beliefs don't apply) and must be bound by public laws. If they don't like the laws, they must get out of the public sphere because the laws governing the public sphere are just and justifiable.
Reacting with horror and anger will just put people's backs up and polarise the debate, and that's never good for democracy.