One of our three users of Max 9 upgraded to build 720. The other two users are still on build 622. When the 720 user goes to open the shared database it asks that the data files be updated to be able to continue. If that user does update the database, will this cause problems for the 622 users who for now can not update to 720?

If the system wants to upgrade files then it is very possible that users running older versions of Maximizer will not be able to access them.

You can get round this by simply updating all users to the same release level, click "Maximizer Live Update" in the Help menu and let it run through.