Since your child was found to have mild allergies to the wheat, milk, and egg whites, remove them from his diet for at least ten days to see if his symptoms clear. Food allergies cause a variety of problems and the only treatment for FOOD allergies is elimination from the diet.
Food allergies are also cumulative meaning each exposure to an allergen produces a stronger response. Additionally, it will take at least ten days of strict avoidance of the allergy foods to see any improvements of symptoms, because allergy responses generally effect a child for 7 to 10 days.
When you eliminate his allergy foods, you do not need to go gluten free, because you will unnecessarily restrict many healthy food choices for your child like oats, rye and barley. Also, be prepared to read every label of every food your child eats. Do not give any foods containing his allergens and learn what I call the code names for wheat (semolina), eggs (albumen), and milk (casein, whey, and many others) so you do not accidentally feed him these foods.
It does not matter how mild the allergy is if it produces symptoms, so strict avoidance and keeping a food diary with notes of symptoms is how you can figure out if these foods are causing your child problems. Also, restricting the suspected allergens for 10 to 14 days and reintroducing them one at a time (i.e. Withhold all allergy foods two weeks and reintroduce milk but not wheat or egg for two more weeks).
Good luck. You are now a food detective. Keep notes of all of your clues and take your child and evidence to a good allergist.
A., mom of 7 year old with severe, multiple food and environmental allergies who's been there and done (still doing!) that :)