Running only Hatchery, JEI and HWYLA. There is no recipe or entry for a chicken analyzer in JEI at all. There are also no recipes or entries for any eggs other than the normal Vanilla chicken egg and the basic Hatchery chicken egg.

I am VERY new to this modpack so it's very possible this is all user error.

I may also have exceptionally stupid chickens because they absolutely REFUSE to read books. I've tried right-clicking, sneak-right-clicking, and also beat a chicken to death with a book and left clicking. I've tried both chicks and adults.

Chicken analyzer and Smart Chickens are part of the Chickens mod. Hatchery is more of a support mod for
Chickens. :)

OH! Then totally user error! Thank you very much for the explanation!

No problem! 😄