Problem with overloading equality and inequality operators

1. Why does it work on simple data types, but not on non-static string members?

class MyClass
{
public:
   char* myString;

   bool operator == (const MyClass& rhs)
   {
      // check for self
     if(this == &rhs)
       return true;
  
     if(myString == nullptr && rhs.myString == nullptr)
       return true;
  
     if(strcmp(myString, rhs.myString) == 0)
       return true;
  
     return false;
   }

    bool operator != (const MyClass& rhs)
    {
      // Always defined in terms of == operator
      return !(*this == rhs);
    }
};