The JLS has this to say (8.9):
The optional class body of an enum constant implicitly defines an anonymous
class declaration (ยง15.9.5) that extends the immediately enclosing enum type. The
class body is governed by the usual rules of anonymous classes; in particular it
cannot contain any constructors.
Now, there is no rule that says that you can't define additional methods (methods that are not in the interface or class that the anonymous class implements or extends) in an anonymous class. This is even though the same argument given that these methods will ultimately not be callable at runtime is true.
Why doesn't this work for enums? I'll leave that for you to decide and report back.