Ah, I missed the part about the fake eggs being missing. I agree on a snake.
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I've noticed that our egg laying has been cut down due to the winter months coming along I think. There wasn't enough sunlight shining into the cop during the day so when I started allowing them to free range in the backyard, I am back up to 4 eggs a day (others aren't old enough to lay yet). During the summer we had some critters digger holes under our coop and coming in to steal the eggs and killed some of our chickens (Racoon we think) so you could have some critters as well, but I would lean more towards the weather and low sunlight. Also, I don't know if this makes a difference but when my girls run out of water or dont get enough during the day, they won't lay until I've refilled their water. Might be a coincidence lol
Found the culprit! Or rather what's left of 'em
It's a white weasel.