Can an enum have abstract methods?

Yes, but you may prefer enum which implements an interface, Look here. I think It looks much better. This is example for abstract method:

public enum Animal {
    CAT {
        public String makeNoise() { return "MEOW!"; }
    },
    DOG {
        public String makeNoise() { return "WOOF!"; }
    };

    public abstract String makeNoise();
}