Brownies without egg
I have used the flax method as detailed in the New York Times with good results.  I make a lot of chocolate chip cookies and I haven't found anyone that can really tell the difference.  Some claim to be able to tell but also say that it is just different and not better or worse.

2 tablespoons ground flax seed mixed with six tablespoons water (or 2 eggs) 

http://dinersjournal.blogs.nytimes.com/2009/09/04/replacing-eggs-with-flax/

Actually for cookies I typically use slightly less that 5 tablespoons of water.  This makes the cookies a bit thicker rather that thing.  Do what you like.