Whole eggs have a long history in cocktails. A lot of these cocktails were "pick me ups" meant to be drunk in the morning. The fat and protein in the egg satiated the appetite while the alcohol helped smooth out the effects of the previous night's drinking (or the aches and cloudiness from a sleepless night).
Most often you will find eggs used in three ways. First is cocktails with eggs which is usually the "pick me up" variety mentioned above. These have a whole egg added to the other ingredients and are shaken with ice rather violently to fully incorporate the egg into the drink. These cocktails are usually strained into a chilled cocktail glass or sometimes over ice.
Second you have batter-based drinks that use eggs to make a sweet, spicy batter. Here you are using a hand mixer as you would when making a cake. The batter is then spooned into a mug with hot water and alcohol - usually rum, whiskey or brandy. The most famous example of this type of drink is the Tom and Jerry.
Last you have egg nogs which are probably the most popular egg drinks today. The first two categories of egg based drinks have faded from the scene. It probably doesn't help that a lot of people freak out about putting a raw egg in a cocktail.
When using eggs be sure to crack it into a separate container first (such as a ramekin) to confirm that the egg is fresh before adding it to the cocktail shaker or bowl. You don't want to have to throw out 2 oz of VSOP Cognac because you cracked a spoiled egg over it.
As far as size, I usually use large eggs here in the U.S. Refer to the egg whites page for more detail in other countries.
So do you have a sad story to tell regarding a spoiled egg? Let's here it so we can all commiserate with drink!