The mayonnaise substitution works both for quick breads using baking powder or soda and for yeast breads. The only difference you'll notice is that the crumb will be closer to the color of white bread than challah.

Is the egg for egg wash or is it for the actual dough. If it is egg wash use milk and if it is for the dough just add a little more water. It really, if only one egg is called for is for is to enrich the dough.

Try substituting 2 tbsp mayonnaise for 1 egg, I find it works well.

There are a few options you can consider. Since the recipe is only calling for 1 egg, you would likely be fine with substituting 1/4 cup applesauce or 1/2 smashed up banana, both of which would add some nice fruitiness to your bread. You could also probably use 1/4 silken tofu as a substitute and I've heard of people grind up flax seed and add a little flax oil to get the consistency of an egg if they're trying to follow a vegan diet. The only other thing is to check the recipe for other leavening as sometime the egg is needed for leavening as well, but with it only calling for 1 egg, that might not be critical here. Hope this helps!