As Bob D has commented, this is very difficult to calculate theoretically. He didn't mention that you need to know the thermal conductivity of the egg's contents – and to make things worse, this conductivity will change as the white and yolk change consistency.
Boiling an egg in a saucepan is, of course, a very wasteful process. Most of the heat supplied goes to making steam or is lost to the surrounding air from the sides of the saucepan.
If you want to know how much heat goes into boiling the egg itself, I think you're better off trying to measure it. A 'steam calorimetry' approach (good nineteenth century Physics) should work... Suspend the egg in a jet of steam at 100 °C (the colourless stuff, not the cloudy stuff that's already starting to condense. Collect the condensate that drips off the egg and, when it has stopped dripping, measure its mass, m, and temperature, $\theta.$
Then heat that has gone into egg = $mL+mc(100°-\theta)$
in which $L$ is the specific latent heat of evaporation of water and $c$ is the specific heat capacity of water. The second term will be much smaller than the first.
[For a well known use of this method, see Joly's steam calorimeter.]