It's my understanding that flax can cause fishy tasting eggs...
Usually adding too many seeds or grains dumbs nutrition down not bulks it up... formulated rations have vitamins, minerals, and amino acids added in that grains and seeds don't.
I bought whole barley at my local feed store... which I sprout as a TREAT, or grow to fodder... it is nutritious *compared* to corn which is very LOW in protein, but it is not complete. They also carry wheat and oats in bulk.
Many seeds are very high in fat and should be fed sparingly. If you want higher nutrition.. feed higher nutrient feed like unmedicated starter, a grower, or a flock raiser... And don't forget that chickens aren't vegetarian... they
need animal based protein because of amino acids that aren't found in plant sources. These amino acids are added into the vegetarian feeds that are available to us. Any seed fed out will offer max nutrients if at least sprouted... it makes them more BIO available for digestion. Since some things sprout better than others... maybe try a small amount from the regular super market
Whatever you do try... just remember to keep it to under 10% of their total daily ration... so nutrients don't get out of whack.
If your birds are locked in where they can't free range... grass clipping are often enjoyed. And homegrown meal worms can be a lot of fun. Enrichment is different though than nutrition.
Some feed stores can order for you what they don't carry in stock.
Good luck!
