Hijab in Quran and Hadith
Question

Could you please provide me with some quotes from the Hadith and Quran on the importance of hijab for women?
Praise be to Allah.Hijab in the Quran
Verses that have to do with hijab are as follows: 

Allah says (interpretation of the meaning): 

“And tell the believing women to lower their gaze (from looking at forbidden things), and protect their private parts (from illegal sexual acts) and not to show off their adornment except only that which is apparent (like both eyes for necessity to see the way, or outer palms of hands or one eye or dress like veil, gloves, headcover, apron), and to draw their veils all over Juyubihinna (i.e. their bodies, faces, necks and bosoms) and not to reveal their adornment except to their husbands, or their fathers, or their husband’s fathers, or their sons, or their husband’s sons, or their brothers or their brother’s sons, or their sister’s sons, or their (Muslim) women (i.e. their sisters in Islam), or the (female) slaves whom their right hands possess, or old male servants who lack vigour, or small children who have no sense of feminine sex. And let them not stamp their feet so as to reveal what they hide of their adornment. And all of you beg Allah to forgive you all, O believers, that you may be successful.” [al-Nur 24:31] 

Allah says (interpretation of the meaning): 

“And as for women past childbearing who do not expect wedlock, it is no sin on them if they discard their (outer) clothing in such a way as not to show their adornment. But to refrain (i.e. not to discard their outer clothing) is better for them. And Allah is All-Hearer, All-Knower.” [al-Nur 24:60] 
“Women past childbearing” are those who no longer menstruate, so they can no longer get pregnant or bear children. 
We shall see below the words of Hafsah bint Sirin and the way in which she interpreted this verse. 

Allah says (interpretation of the meaning): 

“O Prophet! Tell your wives and your daughters and the women of the believers to draw their cloaks (veils) all over their bodies (i.e. screen themselves completely except the eyes or one eye to see the way). That will be better, that they should be known (as free respectable women) so as not to be annoyed. And Allah is Ever Oft-Forgiving, Most Merciful.” [al-Ahzab 33:59] 

Allah says (interpretation of the meaning): 

“O you who believe! Enter not the Prophet’s houses, unless permission is given to you for a meal, (and then) not (so early as) to wait for its preparation. But when you are invited, enter, and when you have taken your meal, disperse without sitting for a talk. Verily, such (behaviour) annoys the Prophet, and he is shy of (asking) you (to go); but Allah is not shy of (telling you) the truth. And when you ask (his wives) for anything you want, ask them from behind a screen, that is purer for your hearts and for their hearts. And it is not (right) for you that you should annoy Allah’s Messenger, nor that you should ever marry his wives after him (his death). Verily, with Allah that shall be an enormity.” [al-Ahzab 33:53] 
Hadiths about hijab
With regard to the Ahadith : 

It was narrated from Safiyyah bint Shaybah that ‘Aishah (may Allah be pleased with her) used to say: When these words were revealed – “and to draw their veils all over Juyubihinna (i.e. their bodies, faces, necks and bosoms)” – they took their izars (a kind of garment) and tore them from the edges and covered their faces with them. Narrated by al-Bukhari, 4481. 

The following version was narrated by Abu Dawud (4102): 
May Allah have mercy on the Muhajir women. When Allah revealed the words “and to draw their veils all over Juyubihinna (i.e. their bodies, faces, necks and bosoms)”, they tore the thickest of their aprons (a kind of garment) and covered their faces with them. 
Shaykh Muhammad al-Amin al-Shanqiti (may Allah have mercy on him) said: 
“This hadith clearly states that what the Sahabi women mentioned here understood from this verse – “and to draw their veils all over Juyubihinna (i.e. their bodies, faces, necks and bosoms)” – was that they were to cover their faces, and that they tore their garments and covered their faces with them, in obedience to the command of Allah in the verse where He said “and to draw their veils all over Juyubihinna (i.e. their bodies, faces, necks and bosoms)” which meant covering their faces. Thus the fair-minded person will understand that a woman's observing hijab and covering her face in front of men is established in the sahih Sunnah that explains the Book of Allah. ‘Aishah (may Allah be pleased with her) praised those women for hastening to follow the command of Allah given in His Book. It is known that their understanding of the words “and to draw their veils all over Juyubihinna (i.e. their bodies, faces, necks and bosoms)” as meaning covering the face came from the Prophet (peace and blessings of Allah be upon him), because he was there and they asked him about everything that they did not understand about their religion. And Allah says (interpretation of the meaning): 
“And We have also sent down unto you (O Muhammad) the Dhikr [reminder and the advice (i.e. the Quran)], that you may explain clearly to men what is sent down to them, and that they may give thought” [al-Nahl 16:44] 
Ibn Hajar said in Fath al-Bari: There is a report of Ibn Abi Hatim via ‘Abd-Allah ibn ‘Uthman ibn Khaytham from Safiyyah that explains that. This report says: We mentioned the women of Quraysh and their virtues in the presence of ‘Aishah and she said: “The women of Quraysh are good, but by Allah I have never seen any better than the women of the Ansar, or any who believed the Book of Allah more strongly or had more faith in the Revelation. When Surat al-Nur was revealed – “and to draw their veils all over Juyubihinna (i.e. their bodies, faces, necks and bosoms)” – their menfolk came to them and recited to them what had been revealed, and there was not one woman among them who did not go to her apron, and the following morning they prayed wrapped up as if there were crows on their heads. It was also narrated clearly in the report of al-Bukhari narrated above, where we see ‘Aishah (may Allah be pleased with her), who was so knowledgeable and pious, praising them in this manner and stating that she had never seen any women who believed the Book of Allah more strongly or had more faith in the Revelation. This clearly indicates that they understood from this verse – “and to draw their veils all over Juyubihinna (i.e. their bodies, faces, necks and bosoms)” – that it was obligatory to cover their faces and that this stemmed from their belief in the Book of Allah and their faith in the Revelation. It also indicates that women observing hijab in front of men and covering their faces is an act of belief in the Book of Allah and faith in the Revelation. It is very strange indeed that some of those who claim to have knowledge say that there is nothing in the Quran or Sunnah that says that women have to cover their faces in front of non-mahram men, even though the Sahabi women did that in obedience to the command of Allah in His Book, out of faith in the Revelation, and that this meaning is also firmly entrenched in the Sunnah, as in the report from al-Bukhari quoted above. This is among the strongest evidence that all Muslim women are obliged to observe hijab.” (Adwa al-Bayan, 6/594-595) 

‘Aishah narrated that the wives of the Prophet (peace and blessings of Allah be upon him) used to go out at night to al-Manasi’ (well known places in the direction of al-Baqi’) to relieve themselves and ‘Umar used to say to the Prophet (peace and blessings of Allah be upon him), “Let your wives be veiled.” But the Messenger of Allah (peace and blessings of Allah be upon him) did not do that. Then one night Sawdah bint Zam’ah, the wife of the Prophet (peace and blessings of Allah be upon him), went out at ‘Isha time and she was a tall woman. ‘Umar called out to her: “We have recognized you, O Sawdah!” hoping that hijab would be revealed, then Allah revealed the verse of hijab. (Narrated by al-Bukhari, 146; Muslim, 2170) 
Ibn Shihab narrated that Anas said: I am the most knowledgeable of people about hijab. Ubayy ibn Ka’b used to ask me about it. When the Messenger of Allah (peace and blessings of Allah be upon him) married Zaynab bint Jahsh, whom he married in Madinah, he invited the people to a meal after the sun had risen. The Messenger of Allah (peace and blessings of Allah be upon him) sat down and some men sat around him after the people had left, until the Messenger of Allah (peace and blessings of Allah be upon him) stood up and walked a while, and I walked with him, until he reached the door of ‘Aishah’s apartment. Then he thought that they had left so he went back and I went back with him, and they were still sitting there. He went back again, and I went with him, until he reached the door of ‘Aishah’s apartment, then he came back and I came back with him, and they had left. Then he drew a curtain between me and him, and the verse of hijab was revealed. (Al-Bukhari, 5149; Muslim, 1428)
‘Urwah narrated that ‘Aishah said: The Messenger of Allah (peace and blessings of Allah be upon him) used to pray Fajr and the believing women would attend (the prayer) with him, wrapped in their aprons, then they would go back to their houses and no one would recognize them. (Narrated by al-Bukhari, 365; Muslim, 645) 
‘Aishah narrated (may Allah be pleased with her) said: “The riders used to pass by us when we were with the Messenger of Allah (peace and blessings of Allah be upon him) in ihram, and when they drew near to us we would lower our jilbabs from our heads over our faces, then when they had passed we would uncover them again. (Narrated by Abu Dawud, 1833; Ibn Majah, 2935; classed as sahih by Ibn Khuzaymah (4,203) and by al-Albani in Kitab Jilbab al-Marah al-Muslimah) 
Asma bint Abi Bakr said: We used to cover our faces in front of men. (Narrated by Ibn Khuzaymah, 4/203; al-Hakim, 1/624. He classed it as sahih and al-Dhahabi agreed with him. It was also classed as sahih by al-Albani in Jilbab al-Marah al-Muslimah.) 
‘Asim al-Ahwal said: We used to enter upon Hafsah bint Sirin who had put her jilbab thus and covered her face with it, and we would say to her: May Allah have mercy on you. Allah says (interpretation of the meaning): “And as for women past childbearing who do not expect wedlock, it is no sin on them if they discard their (outer) clothing in such a way as not to show their adornment” [al-Nur 24:60]. And she would say to us: What comes after that? We would say: “But to refrain (i.e. not to discard their outer clothing) is better for them”. And she would say: That is confirming the idea of hijab. (Narrated by al-Bayhaqi, 7/93)

And Allah knows best.