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If they're all the same age, it sounds plausible. If you want to prove it, you need to have at least two groups from the same hatch. Feed one group the chicken egg, feed the other normally and see what happens.

Are you sure it's that and not simply that they've matured into their regular egg laying?

Use foam shippers and double box. You will have a much better result. Something like these: https://www.etsy.com/listing/1350367245/18-hole-quail-foam-egg-shipper-sets?gpla=1&gao=1&