Should she wear hijab even though her family will be harmed because of that?

It is a serious crime and a reprehensible action to prevent a
Muslim woman from wearing hijab as enjoined by Allaah and forcing her by law
to uncover her head and face and go out unveiled among people. 

The Muslim has to adhere to the rulings of sharee’ah and
“there is no obedience to any created being if it involves disobedience
towards the Creator.” The Muslim woman’s hijab is one of the duties that she
is obliged to fulfil. The harm that a woman imagines will befall her or her
family may be baseless, or it may not be extreme, and it may be something
that can be put up with. So she has to continue to adhere to correct Islamic
dress. 

If the harm is extreme and certain, or it is most likely that
it will happen, then the woman can take off her hijab in order to protect
her honour and religious commitment, but she has to remain as covered and as
modest as she possibly can. In this case it is not permissible for her to go
out of the house except when absolutely necessary, and there is no
concession allowing her to go out to study or to buy things that someone
else can bring to her. What we mean is that she should only go out for
essential needs that cannot easily be dealt with at home, or for some shar’i
action that she cannot forsake, and so on. 

Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) was
asked: 

In some countries Muslim women may be forced to take off
their hijab and leave their heads uncovered. Is it permissible for them to
do that, knowing that whoever refuses to do that is faced with consequences
such as losing their jobs or being expelled from school? 

The Shaykh replied: 

This calamity which is happening in some countries is one of
the things by means of which a person may be tested. Allaah says
(interpretation of the meaning): 

“Alif-Laam-Meem.

[These letters are one of the miracles of the Qur’aan, and
none but Allaah (Alone) knows their meanings.]

2. Do people think that they will be left alone because
they say: “We believe,” and will not be tested.

3. And We indeed tested those who were before them. And
Allaah will certainly make (it) known (the truth of) those who are true, and
will certainly make (it) known (the falsehood of) those who are liars,
(although Allaah knows all that before putting them to test)”

[al-‘Ankaboot 29:1-3]

What I think is that
Muslim women in these countries should refuse to obey people in charge (the
rulers) in this evil action, because obeying those in authority in doing
something that is wrong is not allowed. Allaah says (interpretation of the
meaning): 

“O you who believe! Obey Allaah and obey the Messenger
(Muhammad), and those of you (Muslims) who are in authority”

[al-Nisa’ 4:59]

If you think about the meaning of this verse, you will note
that Allaah says “Obey Allaah and obey the Messenger (Muhammad), and
those of you (Muslims) who are in authority” – but the verb (obey) is
not repeated in the third case with regard to those who are in authority.
This indicates that obedience to those in authority is secondary to
obedience to Allaah and obedience to His Messenger. If their command is
contrary to obeying Allaah and His Messenger, then they should not be
listened to or obeyed in that which goes against obedience to Allaah and His
Messenger. “There is no obedience to any created being if it involves
disobedience to the Creator.” 

The persecution that a
woman may face in this regard is something that has to be borne with
patience, and she should seek the help of Allaah in being patient. We ask
Allaah to guide those in authority to the truth. I do not think that this
forcing her not to wear hijab can take place unless the woman leaves her
home, but if she stays at home no one can force her, so she can stay in her
house so that she will be safe from this thing. With regard to studies that
will lead to her committing sin, this is not permissible, rather she should
study whatever she needs for her religious and worldly interests. This is
sufficient, and this can usually be done at home. 

As’ilat al-Usrah al-Muslimah, p.
22, 23 

And Allaah knows best.