Jewish law has long recognized the right of workers to form unions. One of the basic rights of a union is the right to strike - without it a union effectively ceases to have any power to protect its workers. Modern day authorities dispute the extent to which Jewish law recognizes this right. Where contractual obligations or even customary amenities are being ignored all agree a strike would be allowed. However striking for improved working conditions such as higher wages or increased benefits, is according to many authorities prohibited. In such a situation a union may only demand binding arbitration.
Rav Moshe Feinstein rules that in a situation where a strike is allowed (which in his opinion does include improved working conditions) union workers may actually prevent, through peaceful means, the hiring of replacement workers, basing his ruling partially on the negative feeling that people have towards "scab" labour.
Workers in essential services i.e. hospitals, public safety are enjoined from going on strike. From a Jewish perspective there is no greater essential service than students having the opportunity to learn. This is especially so regarding the study of Torah. Our Sages declare that "the world exists only because of the speech of students in school"; denying students the right to learn is tantamount to destroying the world. Furthermore Rav Moshe points out that Jewish law requires teachers, in theory, to perform their sacred task gratis. They are paid "opportunity cost" a salary equivalent to what they could have earned had they applied their talents elsewhere. Since they are not paid for teaching per say striking for higher wages would not be allowed. Rav Moshe does say that in situations where teachers are paid 'starvation wages', a situation that unfortunately was not uncommon years back, one may be permitted to strike "for a day or two". In my opinion the prohibition of teacher strikes would apply to extra curricular activities as well, as these are vital for the development of balanced children. While one may not be required to become a teacher one who chooses to do must agree to "sacrifice" some of the rights that others enjoy. Hopefully the rewards make it all worthwhile.