I can't say for sure for chickens, but I know for dogs (and humans) it makes their coats shiney. There's a lot of omega 3's in flax, which is a really good thing to have.
Great idea to plant it, I will have to throw some seeds out!
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The eggs produced would likely be higher in Omega 3. There are some commercial egg farms that use this as a selling point on their cartons to get more money for their eggs!