This video shows you how to get a hardboiled egg in a bottle even though it does not fit through the opening. It is possible to do this without breaking the egg.

You can grease the mouth of the bottle (for example with vegetable oil) to help the egg slide through.

The lit match in the bottle causes the air molecules to vibrate faster and move away from each other, and some molecules escape out of the bottle. The amount of air in the bottle is thus reduced. When the flame goes out and the molecules come closer together, the pressure will be lower than the pressure outside the bottle. This pressure difference causes the egg to be pulled inside the bottle.