Islam does not enjoin any particular dress for men and women as according to Quran the best dress is that of Taqwa (7:26) i.e. of righteoussness/ modesty. Only one verse talks of specifics and that is (24:31) which states what areas should be covered whereas (33:59) does not talk of 'veil' as specific dress rather a garment that would cover a woman's body.
Whenever it is said that 'veil' is not mentioned in Islam, it is assumed that one is proposing that veil/ covering or protection of any sort should be abandoned. A sort of clash between secular and islamists camps. This is not the case at all as no one is suggesting that they be done away with. Clothing and subsequently 'veil' are all cultural issues. Muslims in Indonesia, Pakistan , Saudi Arabia and North Africa dress differently and accordingly to their culture. The Tuareg of West Africa, mainly muslims, have the most distinct symbol of veil called Tagelmust. Its a blue indigo coloured veil and interestingly 'men' use veil and women do not! this does not make them 'non-muslims'.
Quran does not go into specifics of dress because this would be illogical in the diverse multi-cultured world. So if anyone wants to use burqa/ veil they are more than welcome but it should not be imposed upon people as specifics of God which obviously it is not. A dress should conform to norms of decency (as stated in Quran) irrespective of 'what' it actually is.