I’ve been following the arguments about burka bans, and much as I hate these coverings and what they represent (and think countries like Syria are smart to ban them), I think any ban on an article of clothing is not only impossible in the United States, but runs afoul of the First Amendment.
I want to explain why I think the religious arguments — discussed by Martha Nussbaum and refuted here by Massimo Pigliucci — are a bit of a red herring. Any apparel worn for religious reasons is as protected under the First Amendment as any apparel would be if worn for any other reason. It is legally impermissible to ban Nazi uniforms or Klan garb, and it is irrelevant whether a religious nut might claim that his Klan robe (which like the burka, is a covering) reflects his relationship with God.
(A related issue is that deliberately offensive anti-religious attire, like gay drag queens wearing nun’s habits to protest the Catholic Church, would also be similarly constitutionally protected.)
A burka might be worn for religious reasons, but any ban would have to be religion-neutral, or else it would be religious discrimination. Take, for example, the analogy between the burka and human sacrifice:

Nussbaum is a sharp thinker, and arguably one of the most incisive public intellectuals active today. In her essay she introduced the issue in terms of two philosophical traditions concerning the rights of minorities, in particular religious ones. According to John Locke, the law should not penalize religious belief, and should not be discriminatory, that is it should be applied equally to all practitioners of specific religions. Nussbaum’s example is a Supreme Court decision that allows ritual animal sacrifice for religious purposes (yup, you read correctly!) because not allowing it would represent an instance of religious persecution against a specific group (in that case, the Santeria worshippers). I do wonder what the Supreme Court would say if a religious group petitioned to carry out human sacrifices…

The problem with that argument is that no legislature in the United States would limit a ban on human sacrifice to, say, the Aztec religion. Human sacrifice is murder and it is illegal regardless of whether the murderer believes in a particular religion, or no religion.
So, any ban on the burka would not band burkas only for Muslims; it would ban burkas period. No head and face coverings.
Which would mean one of the numerous American cranks (there are libertarians like me who don’t like the government telling me what I can’t wear) would immediately take umbrage, put on a burka, and run out into the street to get arrested, contest the law, and ultimately win.
Now, the state has a right to compel an individual to bare his or her head for a drivers license, and I don’t think they have any business telling businesses they have to hire burka wearers (I would refuse to hire one), but banning the burka?
Forget it. I don’t think it would survive a legal challenge.
Whether that’s a good thing I don’t know. Freedom is not without its costs, and allowing deluded women (or other nutcases) to cover themselves is one of them.