Should the woman quit her job at the company whose French owner deals with an Israeli company?

The basic principle is that it is permissible to deal with Jews and other disbelievers and polytheists, through buying and selling. The Prophet, peace and blessings be upon him, and his companions dealt with the Jews of Medina. The Prophet, peace and blessings be upon him, bought food from a Jew and mortgaged his shield, and Ali ibn Abi Talib, may Allah be pleased with him, worked for a Jew. It is also permissible to deal with belligerent non-Muslims (Harbiyyin), however, one should not sell them anything that would aid them in destroying Muslims, such as equipment and weapons.

As for boycotting Israeli companies to pressure them, whoever refuses to work with them with the intention of harming them and weakening their economy will be rewarded. The legitimacy of boycotting is further confirmed when a large group of Muslims collectively agree upon it, as it then achieves the goal of vexing the disbelievers. As for someone who deals with them, adhering to the basic principle, there is no blame on him. You are permitted to remain in this job, unless your work involves something that aids the belligerent non-Muslims against Muslims, such as weapons.