How can I make cookies with no eggs?
There's a replacement, but it's not an ingredient most people have sitting around -- flax seed.

I use it when I'm cooking for a few of my friends who are vegans.  Grind up some flax seed in a coffee grinder.  For each egg, take 1 TB of flax seed meal, and 3 TB of water.  Stir it up and let it sit for a while -- it'll get kinda slimy.  Use that in place of the eggs.

Obviously, this won't work for anything where the whites and yolks are used separately, but I've used it for years for Christmas cookies -- the only problem is that the dough doesn't hold together quite as well when you're dealing with refrigerator cookies.

update: it's also a good idea to make friends with your neighbors.  I've known the folks living in one of the houses behind me since high school ... we've been known to call each other up when we're short on sugar / flower / eggs / milk / vanilla, etc.