I read here about certain Jewish sects prescribing face veil. Is there, in fact, a rule requiring or encouraging covering the faces of women in presence of unrelated men? If so, what is the source for this rule?
This sect is referred to by many today with a mixture of amusement, frustration, and annoyance. Most will tell you that they are bringing the rules of modesty to an unprecedented extreme for which there is no source nor Jewish tradition.
However, a while ago on the Seforim Blog, Marc Shapiro addressed this sect (which he calls the "Jewish Taliban Women"), and brings many sources that recommend or praise the use of such full-body veil (burqa-like) coverings. I will not reproduce his words here, but his sources include the Jerusalem Talmud (as interpreted by its commentaries), R' Baruch Epstein (1860-1941), and R' Joseph Messas (1892-1974). Additionally, Shapiro lists some contemporary rabbis who have written/spoken in favor of the shawls worn by these women.
I am pretty sure RaMbaM said for a girl to cover her hair at age 3 because that is when she is sent to the market and is in the public's view. Once married she needs to put a rodid on top of the mitpahath. (Hilchos Ishus Perek 13 & 24)
I also have pics to display to show as examples, but that will all be tommorow.