"I do not work on Saturdays and Sundays."
If you look at a literal translation it would say, "Saturday and Sunday work not" with Saturday and Sunday being the topic of the sentence and why は is used.
I believe に is confusing you as it would literally mean working on our in those days as if they were physical objects. Like working on a car.
At least that's how I take it.
Actually はたらくfits a little better in this exercise than しごとをする does. People are treating the はたらく like a verb and しごとas a noum, which is partially correct, except for a few nuances and grammar rules. According to http://4friendsnihongo.com/blog/2017/03/12/009_work/ ,hataraku is often used for describing typical work schedule. I hope it helps, the link above is a little long but worth taking a look. Correct me if i said something wrong, I'm still learning.
仕事をする can be used there, like i said in the previous answer. It can
t mean "dont have work", though. Thats where ありません takes place, or you can use ならばなりません with the verb in the negative form to indicate an action is not needed.
Still, I think both options have a different nuance from what is stated in this exercise.