When you wake up, your body has essentially been fasting for the past 6-9 hours and has begun turning to your muscles for fuel, so you obviously want to stop this process ASAP. Eating eggs or other whole-food proteins, like chicken or dairy, will take far too long to digest and stop the attack on your muscles. The only thing that will work is a whey protein shake, as whey is the fastest digesting form of protein, and in less than 20 minutes, its amino acids will start to reach your muscles. Adding some extra branched-chain amino acids (BCAAs) to the shake will further help stop catabolism (muscle breakdown) and turn on muscle protein synthesis.
Also, consider eating a piece of fruit with your protein shake. When you fast all night, your liver uses up its stores of glycogen (the storage form of glucose), which signals the body to use amino acids from your muscles for fuel. The small amount of glucose also doesn't cause a huge blood glucose spike just enough to raise insulin a bit, which also helps to signal the body to stop breaking down muscle.