There are two common methods to make an egg float. One is chemcial method and the other is physical method.
The first method, already given in the previous post, is to add vinegar into the water. The shell of the egg mainly composes of calcium carbonate. An acid (vinegar) could react with calcium carbonate to give out carbon dioxide. Hence, bubbles of carbon dioxide gas will be formed on the egg shell. This gives a buoyance force floating up the egg.
The second method is to add salt into water. By Archimedeus' Principle, the upthrust acting on the egg equals the weight of water displaced.
That is, upthrust = V.d.g
where V is the volume of water displaced, and which equals the volume of the egg,
d is the density of water
g is the acceleration due to gravity
Therefore, by adding salt to water, the water density d will increase. This leads to the increase of upthrust.
If the density of "salt-water" is high enough, the upthrust will be equal to the weight of the egg. The egg then begins to float.