Hi!
No : they won't.
In fact, some of my smallest chicken hens even use my biggest rosemaries as perches...!
But they don't damage them.
Thyme!
Your chickens won't eat your thyme, but still : do think to protect the base so they can't scratch at the roots!!
Advantages are : thyme is really easy to sow (it tends to do it by itself), is pretty to look at, and smells really good...!
Besides, you can plant it literally everywhere...
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...When you say "hang bunches of them in the chicken coop"... you mean in containers - am I understanding you right?
If so, you can plant Mint in pots : your chickens won't eat it, and you can use it as a natural anti-lice/mites...!!
(But do be careful about its flowers : I don't know for US, but here in Auvergne, my mint does sow itself in my yard every year IF I don't cut the flowers... and it is then quick to be invasive!!)
I KNOW chickens eat Basil...
I sadly can't grow it anymore partly because of it.
(That's a shame : fresh Basil is delicious in salads!)
...I regularly mix dried, powdered Oregano to my chickens' food, because it is good for them.
But I really don't know if they would eat the plant itself?
Be careful : chickens LOVE eating basil like crazy!