Why do ugly, mixed breed ducks & chickens end up healthier & live longer than beautiful show birds?

Well, I'm no expert on poultry or fowl but I would guess it's the same reason why mixed breed dogs are heartier and healthier than purebred dogs. Purebreeds are continuously inbred so any health defects are magnified with each breeding to another purebreed with the same health issue. Generally each breed will have certain problems that are typical for the breed that are only perpetuated by the same type of breed (with the same defect) being mated again and again.

Mixed breeds don't have this problem since each parent most likely will have a different health issue that cannot be carried on since the healthier genes will be dominant over each recessive sick gene.

I think ;)

This is true with most types of animals. The gene pool is more limited in a purebred animal and eventually they become less hardy. A hybrid on the other hand gets input from both breeds and they do tend to be more hardy and disease resistant than either of the pure breeds that they came from. You will see the same thing in populations of people. The small closed societies, such as the Amish communities, tend to have more inbreeding, resulting in higher levels of retardation and birth defects.

Also. if you subscribe to Murphy's Law, the more the bird costs, the harder it will be to keep alive! The mutts are a dime a dozen and the purebred showbirds can be several hundred dollars each, so which one will live longer?

Selective breeding causes more genetic problems. It's called hybrid vigor.

to much inbreeding