Array of Strings and Callbacks
Several issue with this code. First, use public/protected/private for classes. Second, use a const char* instead of String or even better str_P. It is the usage of String that is giving the compile error. Third, the Test::TestCallback function reference is not possible. It is a member function. To correct that either have an instance or define the callback function as static. 

You could define an abstract MenuItem class with the string and a virtual member function. A Menu class would hold a collection of MenuItems. All the static data should be put in program memory. 

There is a Menu system in Cosa. Please see the example sketch. This implementation follows the MVC design pattern.  

Cheers!