I don't know how the thyroid functions in chickens, but I know about humans. I have taken Levothyroxine since I was about 12. My thyroid is about half the size of a normal one and produces almost nothing. That dose (.3 gm) is insanely high, I take anywhere from .125-.2mg. In humans thyroid imbalance can cause issues with everything from weight, temperature, reproductive changes, digestive and heart. I would not be at all surprised if a thyroid imbalance caused reproductive issues in chickens too.Thyroxine and shell quality... Not having a lot of luck other than that one little quote. That source also doesn't say HOW much to give, though another study I read used 0.3 gms. But they weren't testing for shell quality.
One good thing... Thyroxine seems to be easy to get.
https://www.petmd.com/pet-medication/thyro-tabs-canine-levothyroxine-sodium
"Levothyroxine ... can also be used to treat hypothyroidism in birds and tortoises....
"Levothyroxine is a synthetic form of a naturally occurring thyroid hormone called thyroxine."
Well. That was fun. Don't know if it will lead me anywhere. If it does, I'll let you know.