You cannot separate religion from government. Insofar as government can be concerned, God is incidental; the true nature of religion is as a moral system evolved to manage societies. The appeal to a higher authority is useful in that it helps extend that system to a diversity of new member of those societies, but the difference between your instinctive aversion to someone shouting racist comments in public and a strict Muslim’s aversion to an uncovered woman is simply in the mechanism by which those moral codes are passed on, justified, and enforced; and many moral rules that have become so ingrained in society that they are held by atheists every bit as much as they are by believers are derived from the philosophy of a religious society. In short, government must draw on some moral code, and there is no fundamental distinction between theistic and atheistic morality that would allow use to prevent religion from wielding a significant influence on policy.
From a slightly different angle, religion is not some activity cordoned off from a person’s life- not in most cases, at least- and it would be impossible for a person to ignore its teachings whenever they step into the world of politics. Religion is an entire system of thought, a worldview that cannot help but influence areas well beyond its theoretical bounds.
As an atheist, I hate to have to say all of this, as I do believe that religious teachings are almost entirely bunk, and I despise most of the political and social decisions that have resulted from them. However, the fact remains that as a democratic nation populated largely by believers, it is nonsensical to expect religion to stay out of our government. However, that is not to say that government should make an official allegiance to religion. By the concepts above, to ally with all religions is to ally with all moral thought, and is thus meaningless; and far worse, to ally with one religion, even momentarily, is to alienate the diversity so valuable to this nation and this world. Thus, the government should do as it always has, at least nominally, in following the morality of the moment, with fidelity only to that popular sentiment and not to its origins, religious or otherwise.