To add just a little to jake's excellent explanation of the relevant Halachic rules:
It's important to remember that Jews are required to adhere to many commandments and restrictions that God set up to apply specifically to Jews. Roughly speaking, these particularistic commandments are not statements of universal ethics or morality; they're statements of God's particular mission for the Jews.
For example, regarding the Sabbath restrictions, God tells Moses (Exodus 31:13-17):
"And you, speak to the children of Israel and say: 'Only keep My Sabbaths! For it is a sign between Me and you for your generations, to know that I, the Lord, make you holy.
Therefore, keep the Sabbath, for it is a sacred thing for you. Those who desecrate it shall be put to death, for whoever performs work on it, that soul will be cut off from the midst of its people.
Six days work may be done, but on the seventh day is a Sabbath of complete rest, holy to the Lord; whoever performs work on the Sabbath day shall be put to death.'
Thus shall the children of Israel observe the Sabbath, to make the Sabbath throughout their generations as an everlasting covenant.
Between Me and the children of Israel, it is forever a sign that [in] six days The Lord created the heaven and the earth, and on the seventh day He ceased and rested."
It's clear from this language that the Sabbath represents a particular covenant between God and the Jewish people, which the Jews observe as a perpetual testimony to God as Creator. For whatever reason, God chose to only require the Jews to fill this role, not all of humanity. So, universally speaking, is doing work on the Sabbath day a bad thing? No. If you're subject to a covenant that forbids it, is doing it a bad thing then? Yes.
So, in these terms, your question boils down to "Is a Jew asking a gentile to perform acts that are forbidden for the Jew on the Sabbath compatible with the Jews' Sabbath covenant or not?" Intrinsically, there's no problem; the Jew's personal restraint in this situation fulfills the covenant. However, as jake explained in detail, the Rabbis eventually legislated additional restrictions so that Jews won't see their work getting done on the Sabbath and forget about their special role.