Do you mean a head scarf or a veil? I believe that Muslim women are permitted to wear headscarves, or burquas or a niqab (a veil) as to conceal themselves from men other than her relatives.
Unless you mean another type of headscarf which has nothing to do with Islam then maybe it will be seen as an informality. And in some countries, the veils have been made illegal.
No - the hijab symbolizes that the woman is forbidden to men ... she should wear it when in the company of any man that is able to marry her (so father, brother, etc - she doesn't need to wear it).