Making Arnel's Originals Buckwheat Bread Mix without egg is easiest using flax gel, which is something you make, not buy. Here's how:
Take 1 Tablespoon of flax meal and mix it with 3 Tablespoons of boiling water and set it aside. (It's a good idea to cover it while gelatinizing.) In 5 to 10 minutes, it will gelatinize and have the consistency of a slippery egg white.
Follow the directions on Arnel's Originals Buckwheat Bread Mix package and when it comes time in the recipe to add an egg, add this flax-gel mixture instead. The loaves of bread will be delicious and moist but when it cools, the eggless loaf will be slightly shorter in height as the loaves with an egg. Below are two loaves made with a very sophisticated bread machine from Japan called Zojirushi, one with an egg (on the right) and one without an egg (on the left).
For another EGG FREE alternative, see the post on this blog for Sour Dough Bread using Arnel's Originals Buckwheat Bread Mix