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.
Which verses of the Qur'an recommend hijab for men / women and what is its justification? The Qur'an was the first address to men regarding hijab. God Almighty says in the verse: “Say to the believers that they lower their gaze and guard their private parts. That is purer for them. Indeed, God is all-knowing.” Tell men of faith to see, to humble themselves, and to be chaste, for this is purer for them, for God is aware of what they do, so that men, by controlling their gaze, block the way to any encouragement to present women in society. . (1) Verse 31 of Surah Noor is known as the verse of hijab. It is said that with the revelation of this verse, hijab became obligatory on women. And say to the believing lower their gaze and preserve representative vaginally and Ebdane adornments except what appeared to them and beating their necks on bosoms and Ebdane adornments except to their husbands or fathers ... and Aadharbn Borgelhn to know what they hide of their adornment )31 ( And say to the believing women: "Shut up your eyes and purify yourselves, and do not reveal your ornaments except what is evident from it." And they should put their headscarves around their necks, and not reveal their ornaments except to their husbands or fathers ...; And do not stamp their feet to reveal what is hidden from their adornment. It does not mean to reveal the ornament in the phrase "Do not display their adornment" It is not showing the parts of the body that are usually adorned [such as the ears and neck], not showing the jewelry alone; Because showing jewelry alone is not forbidden like earrings. "Khumr" is also the plural of khumar, meaning the covering with which women cover their heads. In the phrase “And let them strike their pockets with their wine.” (They should wear a headscarf around their necks.) Women are instructed to wear headscarves over their chests so that their hair, corners and neck are not visible. Verse 59 of Surah Al-Ahzab also speaks of hijab, which is known as verse "Jilbab". In it, women are asked to cover themselves with a robe. The lexicographers have defined the jilbab as a garment larger than a scarf and smaller than a robe, which women wear over their heads and cover up to their breasts. (2)