[masaegu]

That is really simple. 'Mashou' is added to real verbs, and 'shimashou' is added to verbs that end with 'suru'. The verbs that end in 'suru' , strictly speaking, are actually in the form of 'noun + suru'.

In your sample phrases, 'owaru (= to finish)' is a real verb. So you add 'mashou'. But 'kyuukeisuru' is in the "noun + suru" form. 'Kyuukei' means a rest or break, which is originally a noun. We just add 'suru (= to do)' to the noun 'kyuukei' and treat it like a verb.

Hope this helps. You asked a good question!