All perfect praise be to Allah, The Lord of the Worlds. I testify that none is worthy of worship except Allah and that Muhammad is His slave and messenger.
It is not permissible for a Muslim woman to give up wearing Hijab whether she is in a Muslim country or not. She should, rather, adhere to the rulings of Sharia more strictly in the non-Muslim countries than the Islamic countries. Allah Says (interpretation of 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.} [Quran 33:59].
No doubt, exposing beauty in front of Kuffar is more offensive than exposing in front of a Muslim. So, one may go out of her house to fulfill her needs in complete Hijab.
However, if one cannot wear Hijab for peculiar circumstances, then she should avoid going out of her home except when she is compelled to go out to fulfill her necessities. In this case, when she goes out she should return to her home as soon as she completes the necessity for which she went out. Know that necessity is estimated only to the extent of its direness. So, if she is not compelled to go out she should remain in her home because protecting one's religion is more desirable than leaving Hijab for uncompulsory activities.
Finally, a Muslim woman should remind herself that adhering to Sharia and following its rulings will bring her easiness sooner or later. Allah Says (interpretation of meaning) in this respect: {And stay in your houses, and do not display yourselves like that of the times of ignorance, and perform As-Salat (IqamatasSalat), and give Zakat and obey Allah and His Messenger. Allah wishes only to remove ArRijs (evil deeds and sins, etc.) from you, O members of the family (of the Prophet, salllallaahu ‘alayhi wa sallam), and to purify you with a thorough purification.} [Quran 33:33].
Allah knows best.