How to change the integer type used by an enum (C++)?

You cannot do this in C++98/03. C++11 does allow you to do it, and without enum class the way everyone else seems to keep telling you:

enum EnumType : uint16_t
{
  Bar,
  Baz,
  Bork,
};

Again, you don’t have to use enum class. Not that it’s a bad idea, but you don’t have to.

Does GCC support this feature in its implementation of C++11?

Which version of GCC? It looks like GCC 4.4 added this functionality, but you should probably look to more recent versions, just for the sake of stability.