What verses of the Qur’an urge for the veiling, or covering of women? Do they call any reason for such? Do any verses urge for veiling of men?
Sticking only to the Qur'an (and ignoring ahadith which affect things like men wearing gold and silk and impersonating men/women), I list the relevant ayat below. (There's a lot of debate about what "veiling" means, I want to avoid this.)
The reason for women wearing hijab is given as "...they will be known and not be abused..." in Qur'an 33:59.
No verse urges the veiling of men.
And tell the believing women to reduce [some] of their vision and guard their private parts and not expose their adornment except that which [necessarily] appears thereof and to wrap [a portion of] their headcovers over their chests and not expose their adornment except to their husbands, their fathers, their husbands' fathers, their sons, their husbands' sons, their brothers, their brothers' sons, their sisters' sons, their women, that which their right hands possess, or those male attendants having no physical desire, or children who are not yet aware of the private aspects of women. And let them not stamp their feet to make known what they conceal of their adornment. And turn to Allah in repentance, all of you, O believers, that you might succeed.
O Prophet, tell your wives and your daughters and the women of the believers to bring down over themselves [part] of their outer garments. That is more suitable that they will be known and not be abused. And ever is Allah Forgiving and Merciful.
And women of post-menstrual age who have no desire for marriage - there is no blame upon them for putting aside their outer garments [but] not displaying adornment. But to modestly refrain [from that] is better for them. And Allah is Hearing and Knowing.
There is no blame upon women concerning their fathers or their sons or their brothers or their brothers' sons or their sisters' sons or their women or those their right hands possess. And fear Allah . Indeed Allah is ever, over all things, Witness.
Both men and women
O children of Adam, We have bestowed upon you clothing to conceal your private parts and as adornment. But the clothing of righteousness - that is best. That is from the signs of Allah that perhaps they will remember.
O children of Adam, take your adornment at every masjid, and eat and drink, but be not excessive. Indeed, He likes not those who commit excess.
I believe this list is exhaustive, but correct me if I'm wrong.