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What feed are you feeding? What scratch are you using? In what amounts? What are you diluting the buttermilk with? In what ratio?
You said you added it to their feed to keep them from pecking at each other. Has pecking been a problem? What size is your run? What size is your coop? How many chickens do you have (and what type)?
You don't have to bake and grind the egg shells; just rinse them off, crush them up and give them back to them.
There are probably as many reasons for egg eating/breaking as there are chickens doing it -- or at least close. It can be learned -- I've had hens learn to eat eggs after watching me feed them to the hogs and getting a taste of them there. It can be as a result of boredom -- if they are penned and lack environmental stimulus eggs can become the only interesting thing to peck at. It can be a result of nutritional deficiencies -- such as if a fortified diet is cut with too many treats that result in an imbalance.
I have eleven hens in a 7' X 8' coop. I have 7 Red Stars, 2 Black Stars, 1 Easter egger & 1 unknown. I am feeding them Purina Layena & the the scratch is Purina also. I keep their feeder 2/3's full & they get plenty of water..... I dilute the Buttermilk 50/50 with water. I have a few hens that are missing their feathers at the base of their backs where the tail feathers start.
If I'm not mistaken Buttermilk is only 29% CP to begin with. When you cut it 50/50 with water the resulting mix is 14.5% CP. At that point it fails to be a "protein supplement". Between that and the scratch -- which is likely around 9-12% CP. Their diet is CP deficient. And that's before taking into consideration the actual make-up of those proteins. This may or may not be at the root of your problem.
I would agree with Boyd that space issues may also be at play here though. Being cooped up in about 5 sq. ft. each all day every day is bound to lead to some problems sooner or later.
There are home remedies for egg pecking/eating. Some use wooden eggs, some fill an egg with hot sauce, etc, etc. I, personally, would cull the egg eater/pecker now and be done with it and then remedy the other situations to hopefully prevent problems with others in the future.