My understanding is that in a time of gezeiras ha-shmad, i.e. when goyim are trying to force Jews to violate the Torah just for the sake of violating the Torah and giving up Judaism, then it is required to give up one’s life rather than violate any mitzva or halacha however minor it may seem (the example given is if goyim wear red shoelaces and Jews wear black shoelaces, then in a time of gezeiras ha-shmad a Jew is required to give up his life rather than wear red shoelaces). If, however, the goy needs the labor of a Jewish prisoner or slave-laborer on Shabbos for his own benefit, but doesn’t otherwise care if the Jew keeps Shabbos on his own time or not, and it’s not a time of gezeiras ha-shmad, then the Jew is allowed to work on Shabbos. But the halachos (Yoreh Deah 157) are more intricate than I’ve been able to present them here so don’t do anything without learning them carefully and asking a Rav. May Hashem protect all of us from being subjected to such trials.
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