Q. Should non-Jews have to keep the Sabbath?
Since you assume that the Sabbath applies in the first instance to Jews, I will leave out the reasons why Jews should observe the day carefully and strictly.
Concerning non-Jews, there is no need for them to light candles on Friday night or to say Hebrew prayers, but they would definitely benefit from a weekly withdrawal from the pace, pressure and demands of the weekdays, using the time instead for personal and family renewal and cultural activity.
For Jews this all happens on Saturday. Non-Jews can choose Sunday or any other day that suits them.