The essay exam purpose is to let examiners know it is you who really did the assignment. If you mention what methods, architecture, GUI design etc then you will be fine.
Example if you did your networking using RMI and the essay you mention Sockets then clearly the 2 persons are not the same. Or if you use a service layer and did the locking in it but in the essay you mention the client can directly call lock() and unlock() methods... then you are screwed.
So no you don't need to write code snippets but do or should mention "specific" methods or classes you created for such and such purpose.