When we burn something, a chemical reaction is caused between the burning material and oxygen in the air. What we end up with is carbon as a solid product. When we burn the eggshell from every side and make it totally pitch black, a thin coat of carbon covers the whole surface of the eggshell. That’s why the egg becomes black because carbon is black.
In the carbon coating, some air is stuck. And won’t get wet. When the egg is submerged into the water, light rays hit it through the water. When the light rays reflect back from the egg it first have to go through the water and then the air before it can hit our eyeball i.e. retina.
This change causes total internal reflection of the light. When the light hits the surface of the egg it actually hits the air stuck with the carbon. This causes the total internal reflection, which makes the egg look like a silver egg.