Religious duty, as far as I know. However, in Islam, not many things are directly decreed in the Quran. This means that most of the "rules" are based on hadith (things the Prophet reportedly said). Of the various learned people who assembled and interpreted these hadith, 4 wrote what became canonical law books. So in Islam there are 4 legal "schools" that people can follow. Those law books explain both religious and "worldly" law, i.e. when and how to pray, for example, as well as the law governing inheritance.
Anyway, I have not read the law books, so I don't know what they say about hijab, but the Muslims I have spoken to consider it a religious duty.
"Addendum: I was also interested in the statement that “Muslim leaders concur that Islam does not require a woman to hide her face.”"
As far as I know, the above is true. All Muslims I have asked have told me that they consider the hijab (the head scarf) to be a duty, but not the niqab (the full veil). However, I cannot give you any sources to support that.