More than you can't, you don't want to. Every enum, even Java's fairly cool enums, is code oriented.
It's exactly the same as a collection, but with an enum you tend to write duplicate code whenever you encounter it--with a collection you are more likely to write a loop.
I suggest you create a class with a private constructor and have it create the instances of itself, then provide a getInstance(String) to retrieve an instance. This is like the old typesafe enum pattern.
In the long run, however, it's better if you can manage to get enough intelligence into that class where you aren't ever differentiating on a specific instance--going from the "Enum" way of doing it:
if(myEnum.stringValue.equals("EnumTarget"))
executeCode();
To the OO way of doing it:
myEnumLikeObject.executeCode();
Moving the code you wish into the "enum"--preferably delegating directly to a contained object that is instantiated and set into the "enum" at creation time.