Ummm..... Anonymous classes are classes that have no specified name. The only time I use them is when I need to impliment an interface where the definition is rather short. This does not actually override any methods. You might be able to do that by extending a class (I don't know if you can), but then you're extending the class, defeating the point of discussing it.
To my knowledge, there is no way to override a method without extending a class. There are classes that act only as they are told to by certain interfaces. For exmaple, a JButton will normally only react to a click by calling ActionListener.actionPerformed on the ActionListeners added to it. You might think of this as an override, but it really isn't. It's closer to altering the JButton's underlying model in the preferred manner. Those interfaces can be implemented by anonymous classes, but you're not performing a "dirty override."
The only even semi-dirty way to work with Java that I know of is through the reflection package, but that's really just a work around sort of thing, not entirely dirty, and you can't use it to override methods.