Christians are a people of one belief, one loyalty and one agenda. Their faith unites them in solidarity. You might want to learn more about them before flippantly attempting to turn my words against me like a belligerent tween.
It would be beautiful if this was genuinely true, but Christianity has had factions for centuries, because even though a unified belief in Christ should make us overcome the differences between us to serve God in solidarity, it hasn't. Consider the many wars, tensions, and conflict waged between Christians with one another over the ages. On a smaller scale, consider the feuds and bickering just on this forum between Christians regarding hermeneutical interpretations and oh so many other matters. Christianity was the dominant religion at the time of the Civil War, and soldiers who were brothers in Christ and in some circumstances actual brothers savagely fought against one another. One of the fracturing issues actually was religion because many anti-abolitionists cited the multitude of verses from the Bible regarding slavery as endorsement for it.
Look at what the Christian militias are doing in Central African Republic right now, and how the tenets of the Bible have been secondary to their personal agendas. Look at the feuds between Protestants and Catholics in Northern Ireland.
I'm grateful to anyone who has the ability to serve in the military and willingly chooses to do so because it is indicative of a powerful loyalty to our country. When you have someone serving for your freedom their competence should override your feelings about their religious beliefs. Many atheists are moral, upstanding, intelligent, brave, and precisely what our country needs. What they personally believe is none of our concern so long as they perform the duties of their position with devotion to the best of their abilities.
Oh, and there's speculation as to whether the first six presidents were sworn into office on the Bible or a book of laws. We know John Quincy Adams did not use the Bible. Many of the references to God in our Pledge of Allegiance and on our currency were added much later on for propagandistic reasons. You're correct that many politicians often say, "God Bless America" when in public, especially while campaigning. I don't know how many are sincere when they say it.
So yes, of course I think atheists and anyone else who meets the standards for service and chooses to enlist should be permitted to do so.