So far as I know, you can get it two ways - one of which is getting any other type of egg. In this method, you hunt eggs normally but 'get lucky' when picking up any other egg and it gives you a 'you found egg #101" message instead of the normal number.

Alternatively, if you collect all 100 'normal' eggs, and then go to the MoD fountain, you will get 101 if you throw some gralats in. This also works if you do not have all 100 eggs, but you simply get one you do not have. At least, this has worked in previous years.

Players could get two 101's by getting one the normal way, collecting all 100 varieties, trading it in (or to another player) and then going to the fountain to get it a second time.