In my opinion no where in the Quran God says women has to wear hijab. Moreover God also asks men to protect their chastity. Chastity is as much important for men as it is for women. Let's see the following verses:
"Say to the believing men that they cast down their looks and guard their private parts; that is purer for them; surely God is Aware of what they do." [Quran 24:30]
People keep on talking about women's chastity but God wants men also to protect their purity. Chastity is not a feminine thing. It's a valuable thing for both men and women.
The next verse talks about women:
"And tell the believing women to lower their gaze and be modest, and to display of their adornment only that which is apparent, and to draw their veils over their bosoms, and not to reveal their adornment save to their own husbands or fathers or husbands' fathers, or their sons or their husbands' sons, or their brothers or their brothers' sons or sisters' sons, or their women, or their slaves, or male attendants who lack vigor, or children who know naught of women's nakedness. And let them not stamp their feet so as to reveal what they hide of their adornment. And turn unto God together, O believers, in order that ye may succeed." [Quran 24:31]
Here God is asking women to cover their chest (breast) but He doesn't say women have to cover their face or head. The hijab is a cultural dress of the Arabs and it's NOT prescribed anywhere in the Quran.
In general, men and women both are asked to guard their modesty and refrain from showing arrogance.