First, it must be emphasized that non Jews are in no way commanded to rest on Shabbat. On the contrary, it is "a Sign between Me and the Children of Israel". (I have discussed elsewhere the issue of potential converts observing Shabbat,as well as Noachides who wish to observe mitzvot beyond the Noachide laws). But what about a non Jew, performing a labor for a Jew? Biblically, this is not a problem. The command "that your manservant and maid servant may rest", is universally understood to refer to a slave, not a "servant". Non Jews who became slaves to Jews went through a process similar to conversion; including circumcision and immersion, but with the intent of becoming a slave, not a convert. Nevertheless, a slave had to observe all Torah prohibitions, and, upon being freed, became a Jew in every aspect. They were part of the household. In fact, the slave of a Kohen could eat of the Terumah (heave offering), which would be a great sin if done by an ordinary Israelite. (I have dealt elsewhere with the issue of the morality, or lack thereof, of slavery. Basically, it was a concession to a societal norm, which the Torah put limitations and safeguards on, without forbidding it outright.).Biblically, there is no problem whatsoever of having a non Jew do labor for us on Shabbat. The rabbis, however, forbade many things as a "fence" around the Torah. For example, writing on Shabbat is a Biblical prohibition. The rabbis forbade handling any writing implement, lest we come to write. The handling of said implement would constitute a rabbinic, rather than a Biblical, prohibition. These actions are called "Shevut", designed either to prevent an actual Biblical violation, or else to guard the special feeling of Shabbat. Along these lines, they ruled that instructing a non Jew to do work for us on Shabbat is forbidden, lest we take the idea of work on Shabbat so lightly, that we do it ourselves. Since this rule is completely rabbinic, many exceptions were built into it. For example, a person who is ill, but not dangerously so, can be treated by a non Jew, even if Biblical prohibitions are involved. If the action of the non Jew would only constitute in itself a rabbinic prohibition, then instructing him do perform the action would be a "shevut of a shevut", which is actually completely permitted, although we avoid even doing that without some necessity. (Controversy will often come into this as whether a particular act is a rabbinic or a Biblical prohibition). We generally only do this in cases where either a mitzvah is involved, or discomfort. I have written earlier about the issue of electricity on Shabbat. Some consider electricity as "fire" Biblically, others consider it as fire rabbinically, others as irrelevant. Since there is a doubt, we need to be strict, until such time that a valid Sanhedrin will arise.. But many will be lenient about asking a non Jew to turn on an electric circuit for us. Some, but not all, rabbis permit asking a non Jew, even in the case of a Biblical law, to do work that is needed for a communal mitzvah. If what I ask a non Jew to do will likely, but not necessarily, cause him to work for me on Shabbat, it is permissible. . For example, I can bring my car into the garage of a non Jewish mechanic on Friday afternoon, with the understanding that I will get it back on Monday. Although I know that he will work on it on Shabbat, the fact remains that he could have worked on it on Sunday. That he chose not to, is a question of his convenience and preference, for which I am not responsible. Contrary to popular opinion, something we may not request from a non Jew may not be done by a 'hint" (gee it's dark. It would be nice if someone lit a candle). In fact, in most cases, we must stop a non Jew from performing work for us, unless he is actually doing it for his own benefit. In the early days of Zionism, when there was still pride in all Jewish labor, many recoiled from the idea of needing a non Jew to work for us on Shabbat. Many ingenious inventions were developed and put into place, that would automatically do the necessary work for us on Shabbat. This has mostly gone by the wayside; but was based on ideological, rather than halachic, needs. In all cases of doubt, a rabbi should be consulted. But it must be understood, that not having a non Jew work for us on Shabbat, is not for the purpose of making him somehow "Jewish", but is intended not to undermine our awe of G-d's holy day for us. The exceptions are not "outs" but rather built into the system, in order to meet human needs, with minimal violence done to our transcendent Shabbat experience.