The role of eggs in binding meatloaf is greatly overestimated- just skip it, you'll be fine. I'm with Sam 1148 on skipping the loaf pan- I usually just use a shallow pan (sheet pan or an old, non perforated pizza pan), but I don't put much fat in to begin with.

Try here, You find solution for your problem.
https://cookingrecipeslive.com/meatloaf-without-milk/

I grind a vegetable paste consisting of 1 onion, garlic, and parsley/basil, package of defrosted chopped frozen spinach that I have squeezed ALL of the moisture out of sometimes a small handful of pitted olives and then stir that into some whole wheat matzoh bread crumbs, finely chopped zucchini, red pepper, and 1 tblsp ground flax seed that has been mixed with 3 tblsp water and set to form a kind of paste. AntoniaJames suggestion of using oats is excellent (look for a brand that has been produced on dedicated machinery not used on wheat products.

@stephskitchen.

The addition of the gelatin in stock is also great for meat balls. Make them very nice and moist. I think it was a Cooks Illustrated method---but me and them do not get along after they started charging an expired CC for their website..and sending me unsolicited books with threats of 'payment due turned over to collection'.

My son is egg/dairy allergic, too. I've made meatloaf many times and haven't missed the eggs or milk. I use regular (not lean) ground meat, 1/3 cup of breadcrumbs per pound of meat to bind everything together, and about 1/4 cup of chicken stock (adjusted up or down depending on whether I use fresh or dry breadcrumbs). If you use homemade stock, you get some of the gelatin effect that Sam1148 suggests. The bread doesn't bind quite as well as egg, but there's not a huge difference, especially with homemade stock.

One thing for an All beef meatloaf. Is the addition of 1tsp of unflavored gelatin bloomed in 1/2 cup of beef stock.

Do the other binders also...Sauteed vegies, onions, carrots, with tomato juice instead of ketchup for the meat mix.

Then mix all with bread crumbs and the gelatin broth.

It also replaces some of the moisture/fat of a classic meat loaf made with pork and higher fat ground meats.

Don't cook in a loaf pan...but 'free form' on a wire rack wrapped with foil and drain holes poked in the foil. Coat with the final sauce to bake it on about 10-15 mins before finish time.

If you use a meat that has a bit of fat, you really don't need eggs or any dairy . . . . I also use quick oats instead of bread crumbs to help retain some of the flavorful juices that would otherwise run off, when wheat allergies are a concern. ;o)

I've used oats and sauteed chopped mushrooms!

In one of Julia Child's cookbooks (can't put my hands on it right now) she has a recipe for meat loaf that uses a cup of cooked rice and sauteed onions to mix with the meat. The binder would be the rice. The egg isn't necessary when you add the rice and the seasonings your family enjoys.