I have never used a recipe for icing that included eggs, but to make what I need, I'm going to have to do so. These are the ingredient:

3 cups sugar, divided
3/4 cup milk
1 large egg, lightly beaten
1/2 cup butter or margarine, cut into pieces
1 pinch salt

Part of the recipe states," Cook entire mixture over medium heat, stirring constantly, until candy thermometer registers 235* (about 15-20 minutes)." Now, if I do this properly, will that mean that the egg is to a temp where I don't have to worry about anyone getting ill from it?

Yes.