Egg allergy

Here is a chart with many egg replacements:

http://chefinyou.com/egg-substitutes-cooking/

Keep in mind that egg leaven baked goods and many of the substitutes do not, so you may have to provide more leavening.

Egg can often be eliminated when it is not the key ingredient. In meatballs, breadcrumbs that are moistened should work. Even in muffins or pancakes, for instance, you can skip the eggs -- just replace the liquid at 1/4 cup per egg. You could search, for instance, vegan muffins in the f52 search box. Egg replacer is mostly starch + leavener... look at the ingredients on the box; I haven't been impressed with it, though it's been around and used for years. Also, you can find pages of subs on a similar level as the flax.

A classic vegan substitution is to combine a tablespoon of ground flax seeds with a tablespoon of warm water. This combination has binding properties, but I've only used this method a handful of times, so I'm no expert. I've also read about chia seeds being used, as they develop a gelatinous coating when soaked in water, but I've never tried this.
Xanthan gum is used as a texture modifier and stabilizer in many gluten free baked goods. I think you'd be better off without it. For your purposes, consider trying the flax seed trick, but I wouldn't recommend making anything egg-heavy with it (light-textured cakes, meringues, custards, etc.) as it is mostly a binder.