Project eggs/chickens (from what I have been told) are chickens specifically bred to be something different than that from whence they came; a hen of a certain breed mixed with a rooster of a different breed to make something new. It can also be of the same breed, but different variants within the breed, mixed together...in attempts to breed chicks that are 'new' to the chicken world (or at least new to that particular owner's backyard).
The purposes of project eggs and chickens vary. Some do it because they want to create funky looking chickens lol! (weird color combinations, etc). Other, more purposeful use, is to breed chickens that are more hardy in super cold weather, or other betterment projects designed to push the envelope in what the typical chicken is capable of withstanding in regard to temperatures, and what they are capable of achieving in regard to egg production, meat production, etc.
There is currently a UCLA study ongoing in attempts to uncover the "brains behind the birds" in relation to chickens, and how smart they really are in comparison to how stupid they have been dubbed. I'll try to find the link.
Project Egg or Project Chicken is not to be confused with Egg Project or Chicken Project, which are terms used for ongoing programs geared toward 'chicken awareness' of all different sorts. Usually these programs are political in nature and strive for city ordinances in favor of chicken ownership.
I'm a NEWBIE and the above is what I have been told by various sources off and on the internet as well as books and our veterinarian.