Does the subclass 444 visa stay valid if I leave Australia and come back?

The Special Category visa is tied to your presence in Australia. It is valid from the moment it is granted, usually on arrival, until you depart, and it ceases when you leave the country.

This is not a problem in practice because a fresh 444 is granted automatically the next time you arrive, as long as you are still a New Zealand citizen travelling on a current New Zealand passport. There is no application form or fee to re-enter, but you must keep your passport valid.