If no, what due you use to substitute for cooking?
Eggs (lots of questions)
I'm not vegan or vegetarian anymore, but I can answer your questions.
We have a flock of chickens in our backyard. During the summer months they supply all our eggs and eat all our weeds. In the winter when they stop laying I buy cage-free eggs at the local health food store that are certified humanely raised. They run about $3/doz. We would love to eat even more humanely, but we just can't afford that right now (we eat a lot of eggs).
When I was vegan my favorite egg substitute was flax meal. I even worked out a flax french toast recipe that was pretty good (back when I ate grains). Flax "eggs" can even whip fairly well to hold air for things like pancakes or waffles. If you need your egg substitute to be white (like making a white cake) you can soak whole flax seed and then strain off the "goop" to get your "egg". Here's a good blog post about that.
Our family is vegetarian and I tend toward veganism. I can't eat dairy anymore due to GI issues and I'm also not a huge fan of eggs. However, when pregnant and nursing I feel like I need that sort of protein, so I eat them! We get organic/free-range eggs from our local food coop. But I'm also okay with buying just plain old free-range eggs in a pinch. Phil's Eggs are a good substitute for our usual-- they are cage free and naturally fed. I don't know how "humane" they actually are, though. I think you'd be best off looking for a local supplier if at all possible if that's a concern.
When I bake vegan I use both ener-g egg replacer and the banana/ground flax seed combo.
I do eat eggs now. There were several years I never ate eggs, and a few where I never ate them separately (but did eat them in things, like baked goods, etc). We currently get eggs from one of 2 farmers we know (through CSA type programs) or buy the Costco organic omega-3 eggs. We do stick to organic because dh has found he can't tolerate other non-organic or fresh-farm eggs. We have a local amish company that's in our local grocery stores we also like to buy (they're much closer to the farm fresh eggs for taste & quality, but a lot more expensive).
I always favored the ground flaxseed + water sub for eggs when I didn't eat them. (I made a similar flax-french toast as that above recipe, but it also added applesauce to the mix you'd make for dipping your bread in.).
How humane do you want it?
Cockerels are ground into meal on their first day. Breeds raised for eggs are not used commercially for meat production, so this is the fate of males. The only way around that is to raise them yourself from birds purchased from a neighbor. Large commercial hatcheries also kill the cockerels, as more pullets are ordered. A few hatcheries sell only "straight run", which is batches of unsexed chicks.
I know this is probably beyond the scope of your question, but considering the forum, I wasn't sure how far you needed to examine the issue.....
We had to go egg-free for a while. Flaxseed goo was better than nothing, but not a perfect substitute. For some muffin recipes that I was able to use wheat (for me, I am allergic to eggs) I could sometimes get away with no eggs or egg substitutes whatsoever. Applesauce is good in muffins and pancakes. I've heard prune puree serves a similar purpose.
I don't eat eggs.
When I veganize a baking recipe, I most often use flax slurry (3 tbs warm water 1 tbs ground flax meal), though sometimes I will use applesauce, banana, prunes, blended tofu, or a commercial replacer.
I eat more tofu now than I did when I ate eggs, scrambled or fried with toast for breakast, made into a quiche-like pie, or chopped to make an egg-like filling for sandwiches.
For french toast, 1 cup of milk (I use soy) whisked with 2 tablespoons cornstarch and 1/4 cup chickpea flour makes an excellent batter, just don't taste it raw, the bean flour tastes terrible until it is cooked.