Is it permissible to work for governments hostile to Islam and Muslims (such as the State of Israel) in civil professions like teaching or nursing? And is this considered an act of allegiance?

The fundamental principle is the permissibility of working in lawful professions—such as a doctor or a teacher—even if the state is at war with Muslims. The Fiqh Encyclopedia states the permissibility of trading with belligerents, as long as it does not lead to strengthening them, or involve trading in things prohibited by Islamic law. Residing in non-Muslim lands is conditioned upon the ability to openly practice one's religion; otherwise, it is prohibited, and migration becomes obligatory.