I spent enough time working at a Mc Donalds in high school to tell you that it all depends on which hockey puck you order.
The Mc Muffins
The Mc Muffin family of sandwiches (Egg Mc Muffin, Sausage Mc Muffin, etc), do indeed have one whole medium sized egg. It only fits on the english muffin because of the way it is cooked. The eggs are cooked in a small frame device which handles six eggs at once. You break the eggs into the little individual circles in the frame. Then you place a lid on top, and pour some water into a little hole on top of the lid (they are pretty much poached eggs). This is what gives them that hockey puck shape that is just slightly smaller than the muffin is. These are probably the most labor intensive of all the breakfast sandwichs to make.
The Biscuits
The biscuit family of sandwiches (Sausage Egg Biscuit, Bacon Egg Cheese Biscuit, and the newer "Breakfast Bagels"), do not have an entire egg on them. As a matter of fact, at most locations they do not even have real eggs on them at all. They are made from a carton of egg substitute. These are cooked in groups of three in a rectangular frame without a lid. You start by pouring a tiny amount of the "egg" into each slot. After a short amount of time you fold them up into squares, and put them on the sandwiches. These appear to have more content than the hockey pucks, but it is just fluff, they have the equivalent of about half an egg in them at the most.
Other restaurants
Jack in the Box uses one (or more), whole eggs on most of their breakfast sandwiches. While Burger King uses the "fluff" method to produce round fluffy egg pucks with less than a whole egg. I can't speak for any other restaurants, as those are the only ones I have first hand experience in.