I have a daughter who is now six years old and healthy. At age 1, I learned that she was allergic to milk and eggs, and that explained a lot about why she had been sick so much before that.
When I found out that I couldn't give her milk or eggs, I felt like a truck had run over me. What on earth would I feed this child? But I figured it out, and you will too.
First of all, get real allergy testing so you know exactly what to avoid. Don't just go on your doctor's word for what "probably" is the problem. The "big 6" things that are the most common allergies are eggs, wheat, milk, corn, nuts, and soy, but you need to know exactly what your child's problem substance(s) is/are.
The good news about egg and milk allergies are that they are often outgrown. At 6, my daughter can now drink cow's milk, but still tests as allergic to eggs.
It really isn't hard to live without eggs and milk. It just took some getting used to, and you will get used to it too.
You can give him meats, fruits, and vegetables just like always.
To avoid milk: Soy milk is a great replacement for cow's milk. My daughter liked the taste and chugged it, and it's fine for cooking. I made her pudding with it, and pancakes, and anything else that called for milk. I also bought soy-based yogurt.
To avoid eggs: Don't worry. Google the Food Allergy and Anaphylaxis Network (FAAN) and get a list of egg substitutes. You can't use EggBeaters or other things like that because they are made from real eggs, and that's the allergen. But you can use many different substitutes, including mixtures of water, vinegar, and baking powder, or apricot preserves, or applesauce, or other things. There's a product called EnerG Egg Replacer that contains potato starch and tapioca flour; I have't used it, but I've heard it mentioned many times. I actually bought a box a few weeks ago, but haven't opened it yet. After awhile of cooking without eggs, it's like second nature, and not hard at all. I can even do a pumpkin pie without eggs, and you wouldn't know the difference if I didn't tell you.
To do the pancakes, use Bisquick and the recipe on the side of the box, but instead of an egg, use 1 tablespoon water, 1 tablespoon distilled white vinegar, and 1 teaspoon baking powder. I use a measuring glass I got at Wal-msrt (beside the measuring cups) for a dollar or so. It's 4 ounces, but marked into teaspoons, tablespoons, ounces, and milliliters. Anyway, put the water and vinegar together first, then add the baking powder which will make the whole thing fizzy. This will give rise to breads, cakes, etc.
Incidentally, commercial breads should be okay, as well as Oreos and Chips Ahoy cookies. Stay away from other peoples' homemade cookies, though. Although there are many egg-free cookie recipes you can find online, your friends won't know about them. And look through your cookbooks. You'll be surprised to notice how many dessert recipes don't use eggs at all.
One lucky thing for you is that labeling laws have gone into effect in the past few years. It's now much easier than ever before to check the ingredients in foods.
The main thing is, don't flip out. You will figure this out. Email me for more ideas if you need to. Good luck!