Egg allergies can be challenging, but there are no shortage of other quick protein choices for breakfast. One favourite in our house is cottage cheese with fruit.
For my DS (who is 5'10" and 170# at 14 - hard to believe he used to be "failure to thrive" and tube fed) he has low muscle tone and does find strength and endurance a problem. He has found (if you have a Wii with a balance board) that the Wii Fit yoga has been very helpful for him. He can be terribly self conscious, so this is good for him - he does it at home on his own, and none of his friends know at all. He is also involved in martial arts classes, plays badminton and canoes weekly when the weather will allow.
We found that adding a snack just before bed that was equal parts fat/protein/carb (cheese and crackers, apple slices with peanut butter) made a HUGE difference in how he felt. He woke with more energy, lost a bit of weight, didn't change what he ate in the morning at all, but also didn't wake feeling starved.
He is not terribly overweight, but because of his low tone and significant lordosis, has quite the belly. It is another thing he is self conscious about, but we found that a bit of research about cuts for clothing that don't emphasize his shape and letting him choose his own clothes more helped with his self image, which spilled over into his willingness to keep trying.
I struggled terribly with my weight as a child, and was ridiculed by my parents for it - I swore I would not do the same to him. I did explain to him that because I was obese, he is genetically more likely to have the same problem, and setting habits while he is young will help him in the long term. He walks 12 km a day (6 km each way to/from school by choice) so he seems to "get it". He doesn't remember me at my highest weight, but I am still a fairly big woman, and have lost 150# from my top weight. I struggle to keep it off, but it is only partly body type, and a lot of habit :( Doing things together is the key - he keeps me honest, I help to motivate him.