Much of the liberal dismay over Hobby Lobby (and before that the campaign finance cases) has centered around the question of "corporate personhood" -- whether a company, like an individual, has the various rights protected by the Constitution. I've always thought this was the wrong question. I think everyone agrees that corporations should not have the right to vote, on the one hand. And on the other hand, I think everyone does agree they have the right to the freedom of the press -- shutting down the New York Times and justifying it because "it's a corporation" wouldn't fly with anyone.
So really the question isn't "do corporations have 'individual' rights", it's "what rights do corporations have." But even that, I think, doesn't get at the dismay so many felt over Hobby Lobby. I don't think the problem is necessarily even that Hobby Lobby asserted a freedom of religion claim. There are corporate-religion claims that I can imagine many people finding plausible and worth endorsing -- a kosher butcher suing over state restrictions on meat slaughtering. Another would be the efforts of Jewish-owned businesses to be exempt from Sunday closing laws (efforts that were almost entirely unsuccessful, at least in the courts).
The problem is that Hobby Lobby's decision feels less like "respecting an individual's choice of conscience" and more like "government imposing its religious beliefs onto me." And here's the important point: from the vantage point of its employees, Hobby Lobby bears a much closer resemblance to government than to a fellow citizen.
As a private citizen I might be able to request a religious exemption to, say, wear a beard at work or have Saturdays instead of Sundays off. This has at best an incidental effect on people-not-me. By contrast, if I assert that an entire highway needs to be moved to accommodate my religious sensibility, affecting thousands of other people, I'm going to encounter far more skepticism. We tolerate religious accommodations where they are primarily private matters that have an at most incidental impact on other members of the polity (which I support -- why not?). But Hobby Lobby's decision is hardly private; the primary effect of its decision is felt by its workers. In its ability to seriously and materially impact the lives of thousands of other people, Hobby Lobby is approximating a government far more than a person.
The broader point is that for most people, most of the time, the entity that "governs" their daily lives and conduct is not Congress, its their employer. Your boss largely decides how much you're paid, what benefits you are entitled to, when you can come and go, even what you must wear. There are differences, of course. A business relationship is contractual; I can leave for a new job. But then again, in the formal sense I can move to a new state or country. And in this economy, it is hardly a given that (practically speaking) people can willy-nilly move about jobs or countries. Ultimately, the ability to exit is somewhat more robust in the corporate context. But there are other areas where government has the advantage. I have many routes to influence my government, including voting, campaigning, writing letters, and protesting -- in fact, I have a right to submit grievances. There is typically no functional analogue in the corporate context, and many of the above options are a quick route to being canned for insubordination. Advantage: government.
In any event, the point is not to say whether government or corporations on net do or can deprive us of more liberty. Functionally speaking, most of us most of the time are subjected to corporate, not congressional, governance. That the company plays this role in our lives means people are -- justifiably -- going to mentally slot it in as a ruler rather than a peer when it starts making decisions that impact us. Fundamentally, it's a rejection of the libertarian conceit that economic relations are the free interaction of coequals. Rather, we're talking about what powers "government" -- defined functionally rather than formally -- has over individuals. And when the government tries to say "we're going to obstruct your access to contraception because we find it religiously immoral", people are going to have the same reaction as if an old-school government made the same decision.