Well all believing faiths used to cover, male and female. That's a fact. Even zoroastrian men and women cover! It's one of the oldest monotheistic faith, I studied them before embracing Islam as well, unfortunately much of their teachings have been lost and later perverted (they end up worshipping fire). Their women also wear the veil, their men also wear head coverings like what muslim men and jewish men wear.
Mary was veiled too and is always portrayed as such, whenever movies are made regarding Jesus or Abraham or Moses, the women would also be shown covering their faces at times (as per bible descriptions). The difference is we Muslims uphold these values of modesty today.
There is nothing wrong with the burqa or niqab. In fact I have a few friend's who's wives wear it. In fact one of them is a teacher and she wears it. It's a whole fear mongering and misinformation ploy to make people hate Islam and Muslims today. You don't see anyone criticizing christian nuns, african and middle-eastern jews (not israelis).
Christians have totally abandoned anything linked to the abrahamaic faiths, Jesus or Mary, they follow in the footsteps of the Romans. The Romans are the ones who popularized shaving for instance hence roman catholic priests shave. Who told them not to mary women and be 'celibate'?
In the bible there are plenty of verses referring to veils and what you would call burqa/niqab.
Likewise the jews, have abandoned much relating to God, modesty, God's law, and just carry a label, do whatever their desire tells them.
Yes some people wear it out of culture/tradition, but this is religiously a dress code. Not all muslim women for instance wear it, but some chose to do it. It does not mean that women who cover themselves to this extent are uneducated or incapable to talk for themselves, it's quite the opposite if you listen to some of them be they converts or not converts (as far as muslims go). And yes it is indeed interesting that in parts of Africa the older Jewish traditions are retained you will find women not only wearing hijab but niqab as well

Just go to youtube and look up 'niqab in bible' and 'burqa in bible'. One example of a verse where a women wore the veil, a strange man approaches and she covers her face. Or another example, an evil man rips off the face covering of the woman and rapes her. Stories of the bible for ya.
The whole principle behind hijab, niqab, burqa, etc... is to protect women's dignity not to 'oppress' women as some westerners think. Especially womenizers like sarkozy that made it punishable with jail time to wear niqab in France.
It's ironic that the west talks about liberation of women through exploitation of women. "Cover girl" makeup, more like 'enhance expose girl', it's an ironic society we live in. Where it's 'liberation' to be exploited for one night stands in clubs and bars. To be a 'model' to be a porn 'star', etc... but then women who cover themselves are 'oppressed' for not showing what some pervs wana see.