Eggs are easily substituted on a whole-food, plant-based diet. Whether it's for binding, leavening, or structure, there are several options to choose from. Sometimes, as in pancakes, you can just leave the eggs out without any replacement. Experiment and have fun in your kitchen.
a) commercial powdered egg replacer. To prepare, follow the instructions on the package. Made mostly of starches, rather than whole-food, this option is best for times when you want a lighter result as in cakes or cupcakes.
b) 1 Tablespoon ground flax or chia seeds mixed with 3 Tablespoons warm water, warm applesauce, or left-over water from draining a can of beans (also known as aqua faba). I use the ground flax and water option most of the time. This method is good for binding (especially in gluten-free baking) and adding moisture to baked goods. Best for muffins and quick breads where you want a moist, dense crumb. For examples, see my recipes for Sweet Potato Quick Bread and Sweet Potato Chili with Crusty Cornmeal Topping
c) 4 Tablespoons (1/4 cup, 57 grams, or 2 ounces) blended soft or silken tofu, non-dairy yogurt, or unsweetened apple sauce or other fruit puree such as banana. Apple sauce and tofu are the most neutral tasting. Other fruit purees will add some flavour and sweetness.
d) In gluten-free baking, try adding an extra 2 Tablespoons of starch and 2 Tablespoons of liquid per 1 cup of flour to the recipe for extra binding. Or, add an extra 1 teaspoon baking soda and 1 Tablespoon vinegar to the recipe for extra leavening. I added cooked white rice for extra binding to my recipe for Buckwheat Crepes with Chocolate Orange Sauce.
For egg-based dishes, like quiches and custard-like pies, replace each egg with 4 Tablespoons (1/4 cup, 57 grams, or 2 ounces) of tofu — soft, silken, firm all work. The only differences between them are the water content and how they are set. For savoury dishes, chickpea or other bean flours mixed 1:1 with water are also a fantastic and flavourful egg replacer. The Crustless Mushroom and Tomato Chickpea Tart is an excellent vegan quiche variation. Tofu or bean scrambles are delicious. The Greek Chickpea Scramble can be made with either chickpeas or tofu (I prefer chickpeas). Omlettes and fritattas are like quiches and can be made mixing chickpea flour 1:1 with water. "Egg" salad for sandwiches can be made with soft, silken or firm tofu.
For mayonnaise, replace with hummus, avocado pureed with some lemon juice, nut butter thinned with liquid or a homemade tofu mayonnaise. My recipe for Beet and Napa Cabbage Slaw has a wonderful tofu-based, mayonnaise-like dressing with a hint of citrus and caraway seeds.
To your health and happiness,
Beans and Rice
http://www.farmsanctuary.org/learn/factory-farming/chickens/
https://poultrykeeper.com/rehoming-battery-hens/the-life-of-a-battery-hen/
http://www.humanesociety.org/issues/confinement_farm/facts/cage-free_vs_battery-cage.html
See these links for information about eggs and health:
http://nutritionfacts.org/topics/eggs/
https://www.drmcdougall.com/misc/2005nl/march/050300pueastereggs.htm
http://www.drcarney.com/topics/item/262-unscrambling-the-truth-about-eggs