There is a general prohibition to have a non-Jew perform a forbidden labor (Melacha) on Shabbat, either by telling them directly or by having it done on one’s behalf. This is because the non-Jew is an agent of the Jew and by association, the Jew is complicit in the performance of such a labor. Another reason for this prohibition is because the labor is a mundane, weekday-type of act which is not appropriate on Shabbat, as is alluded to in the verse (Yeshaya 58:13) “Asot Heftzecha” (lit. “Your [mundane] affairs”).
Regarding having non-Jews work on Shabbat, there are three types of workers. The first category is a partner who owns a percentage of a company. Such a person may work on Shabbat because one is doing it not for the Jewish partner’s gain, but rather mainly for one’s own gain. A second category is a contractor, who is a worker who is contracted to do a specific task or project for a set fee. An example of this is a mechanic, who is paid a set amount to perform a specific job on the car. Again, since the non-Jewish contractor is working on one’s own time and work on any day, including Shabbat, by one’s own choice, this would be permitted. In fact, Rabbi Ovadia Yosef (Hazon Ovadia, Shabbat I) rules that one may leave one’s car before Shabbat if one does not specify what day it should be worked on, even if it ends up being repaired on Shabbat. The third category is a salaried employee who gets paid hourly, monthly or the like and who, by working on Shabbat, is specifically being paid for the work performed on Shabbat. An example is a receptionist, who is paid a specific amount for every hour worked. This type of worker relationship is categorically forbidden on Shabbat.
The Shulhan Aruch (Orah Haim 243:1) qualifies the rule regarding a contractor by saying that a contractor may not perform the job in a public place such that onlookers will think that the Jew hired the worker as a salaried employee. For example, an IT specialist may not come into one’s office one Shabbat to work on a specific computer issue as it may appear as though this person is a wage worker who is being paid specifically for the hours worked on Shabbat.
Summary: One may have benefit from the labor of an equity partner or a contractor that was performed on Shabbat. One may not employ a salaried employee for work on Shabbat.