There are two different problems with inbreeding. One is genetic problems. This is what most people think of when they consider line breeding. Extra toes, different colors- problems that are brought about by genes- and for most people they're thinking the recessive genes. Well, except for the rare mutation, inbreeding can only cause genetic problems if the genes for those problems were there to start with. In fact, inbreeding can help you find out what deleterious genes are in your animals. Just remember, all individuals do have some "bad" genes
The second problem is actually worse in the long run. Inbreeding cuts down on the available genes for breeding. Once you lose a gene- it's gone. It also lowers the Major Histocompatability Complex. I'm doing this from a mammalian standpoint as that's how I learned it; a lower MHC increases the risks for lower birth weights, lower litter sizes, increased allergies, decreased lifespan, decreased immune systems. Overall, an increased risk for a poor doer. And it's a risk not a sure thing.
So, inbreeding or no inbreeding? Well- it depends. Are the offspring going to be used for breeding? If so- it just might tell you what genes there are in your lines. The general idea is that you at least keep in contact, if not in control of, all offspring to see what pops up. You'll need to run several on just to see what happens. Ahem- not sure how that part would work out for meat chickens with short lifespans ;-)