People should follow the customs of the culture they immerse themselves in.  Americans take their shoes off before going inside a house or school in Japan.  Japanese shake our hands when greeting us. In Bali it is impolite for women to wear pants or shorts inside a Balinese temple. So courteous westerners dress accordingly and don a sarong.
We should expect Muslims to show the same respect for our culture that we should show theirs. When we communicate, we look people in the eye and show our faces. Anyone unwilling to show this level of professional respect for others is rude and unsuitable for working in any capacity that involves interacting with people face to face.
Just as it would be appropriate for someone to fire a teacher who: spat on students; swore continuously; revealed himself to students, there would be nothing wrong with firing a teacher who insisted that she come to work in a full-face veil.
However, our country should not make a habit of criminalizing rude behavior (see: flag burning). In no way should we criminalize veils. There are plenty of activities which are not illegal which will nonetheless get you fired for doing them in public (see: downloading pornography).
Criminalizing veils or nose rings goes against every ideal our country aspires to. People should be free to make bad, rude choices, and we should be free to ridicule and punish them socially for it.
Anne Applebaum wrote a great article on this over at Slate.