I fed flax for a while last year to chicks that were going to be layers, but quit because I figured I would get more out of the flax seed if I just ate it myself! Why not just skip the step of mixing it into the feed. Plus I cut down on loss of seeds that the chickens didn't eat. Skipping feeding flax to my chickens makes sense economically and effort-wise. Just found out a couple of days ago, though, that most of the Canadian flax seed crops are contaminated with GMOs, so ... not sure what to do now because no one knows exactly how the contamination occured so I need to do some more research before I eat any more of it whether it comes from US or Canada. If Canadian flax can get contaminated without anyone knowing how, then US organic flax could, too.
The eggs would be healthier for you to eat because they would have a good Omega 3 to Omega 6 ratio (which is supposed to be around equal 1:1). However, chickens can make Omega 3 eggs by eating grass, too. Pastured chickens (and ruminants, too) are higher in Omega 3s (have a better Omega 3 to Omega 6 ratio). If you fish, then feeding the offal to the chickens is supposed to be really good for them.
I toss some in the run every once in a while for some variety. I doubt it is enough to make a nutritional difference in the eggs; certainly not more than I would get from eating it straight, as Sparklee already observed.
I did some study last year into supplement flax seed for poultry...unfortunately I cant find the website references at the moment. but what I remember is some sites say poultry dont absorb flax seed omegas that great when feeding whole flax? I think there is greater nutritional rates by grinding the flax before feeding it? I cant recall for certain. maybe someone else knows about this?