There are several instructables that mention using flax seed as an egg substitute. For a comprehensive view of different ways to replace eggs, I recommend this one: https://www.instructables.com/id/Egg-Substitutes/
Flax is a great substitute for egg in dense baked goods, like cookies, brownies, and pancakes. This little instructable shows how to prepare the flax step by step with pictures and video.
Submitted by Ace Monster Toys Hackerspace in Oakland, CA for the Instructables Sponsorship Program
Step 1: Grind the Flax Seeds
You can start with ground Flax Seed meal or whole flax seeds. The whole seeds stay fresh longer so I always have them on hand. Also I use "golden" flax seed because it's less visible when mixed into batter, but the dark seeds will work just as well. If you're using ground flax meal you can skip to the next step.
If you're grinding flax seed from scratch you'll need a spice grinder or a coffee mill. This is a coffee mill that I use for spices and never for coffee, because coffee has a strong flavor. If you already have a coffee grinder and you want to use it for flax seed or other spices: wash out the inside with warm soapy water; rinse to get rid of the soap; then rinse with a little white vinegar. There should only be a faint smell of coffee left.
Once you have a clean dry grinder, simply put the flax seeds in and grind for about 30 seconds. I find it grinds better if there's at least two tablespoons of seed, so I may end up with more than the recipe calls for.
Step 2: The Texture of the Mix
Step 3: Notes on Baking With Flax
Flax meal cookies will dry out faster than cookies made with eggs, so they're best when eaten on the same day they're baked. Don't overbake them! These little chocolate chippers were baked for 12 minutes at 375 degrees. The edges were just beginning to brown when I took them out of the oven; I let them "set up" for five more minutes on the hot baking sheet, then I moved them to a rack to cool. No worries about raw egg in this batter, you can even lick the mixing bowl :)
Eggs in cookies tend to act as an emulsifier - helping the fat and water in the recipe blend smoothly. Flax meal can do the same thing. It won't work as an egg substitute in any recipe that uses beaten egg whites for volume, like meringues. It also won't work in recipes that depend on egg proteins for structure, like custard; or for flavor, like french toast.