Shemuel visited Avin of Turan's house. A non-Jewish [acquaintance of Avin's] came and lit the lamp [on Shabbat]. Shemuel turned his face away [from the light]. When he saw that [the non-Jew] had brought a document and was reading it, [Shemuel] said, "He lit it for his own benefit!" So he turned his face back towards the lamp.
What work did our Sages allow a non-Jew to do for Jews on Shabbat? Very little, as it turns out. As we see in the above story, the third-century Babylonian Sage Shemuel is unwilling to take benefit from the non-Jew's lighting of the lamp, so long as he believes the non-Jew lit it for the benefit of the Jews in the room. Shemuel stands on solid ground here, the Mishnah's principle being that if a non-Jew performed labor for a Jew on Shabbat, it is forbidden to garner benefit.
Non-Jews are not restricted by the mitzvot that apply to Jews. The covenant of Sinai is between God and the people Israel, and Shabbat rest does not extend beyond the boundaries of of this covenantal peoplehood. Our Sages conceived of a number of ways that non-Jews could connect with God, but Shabbat was not among them.
This said, why did our Sages prohibit us from enlisting non-Jews to perform labor for our benefit on Shabbat? One reason might be that Shabbat loses utterly its meaning if any forbidden labor can be accomplished simply by requesting it. But another reason is perhaps deeper: Shabbat is our day of freedom from labor, a remembrance of the Exodus from Pharaoh's house of bondage. In that context, how would it look for Jews to be ordering non-Jews about, when the Jews themselves are free from labor?