MySQL connection class
Don't print anything or call die() from this class because it may breaks caller's code/page. For example, you may want to show customized error page when error occurs, but all text printed by your code will break and stops this. One point is return boolean, as already said. But I personally prefer using exceptions for that (and even more -- I create special excepton which extends standard Exception class. This approach allows to differentiate one exception from another.).
Get rid of chained method calls. Don't call verifyDatabaseConnection() from verifyNullFields(), verifyDatabaseExist() from verifyDatabaseConnection() and so on. Your method should do only one thing as declared in their name.
Rename data member to something more meaningful.
At your opinion: you may replace $this-&gt;user == NULL) check to is_null($this-&gt;user)
Also, when you will need to create similar class for PostgreSQL database (or abother DB) you should extract base class