It is quite common still to come into China on a visitors visa and get it changed to a zclass once you are here. Even schools that have a proper licence do this. Mainly because it is much easier for the school to do once you are here.
If you want a z-class entry visa then you have to send your documents first, and the school has to get a letter, which costs time and money, and then send this back to you. It is more expensive for you and the school and is a pain in the rear.
However, if you want to make sure you 100% legal then demand the school does a z-class entry visa for you. Otherwise ask to speak to other teachers and see if they have a proper visa and how they came into the country.