Hair is part of the body. The Gemara (Sukkah 6a) calls it טפל לבשרו; secondary to the flesh, but clearly a legitimate part of the body, as it needs to be included in tevila (ibid).
As to why @SethJ heard that a married woman's hair is ervah and not an unmarried woman's, the source that hair is ervah is Berachos 24a. The Mordechai there cites a Ra'avya (an opinion echoed by various rishonim) that things which in any particular society people are used to, do not retain the status of ervah, despite the Gemara naming them so. So it's really a case of the chicken and the egg; the single women do not uncover their hair because it isn't ervah; rather it isn't ervah because they uncover their hair.
I fail to see how any of this would have bearing on Hilchos Shabbos.