Unlikely. Farming is a necessary enterprise; and it's very hard to draft laws that target private livestock ownership yet leave commercial operations unaffected.
Big AG has plenty of sway in every state to keep livestock legal.
Rabbits aren't of commercial interest, so there wasn't big business pressure besides the (relatively minor) pet industry.
Just like in Florida, many species of lizards were banned from sale AND to keep existing pets without a permit & habitat inspection 2 years ago for ecological reasons.
I was a bit put out about it because I was looking to get an axanthic (blue) Iguana at the same time, and they can't even survive the winter in the wild as far north in FL as we live. So why ban the whole state instead of just the south? *sigh*
The pet industry just didn't have the juice to do anything about it, despite a full-effort campaign to reduce the scope.
Back to chickens... The clearest route to target private livestock owners is with land use rules enacted by counties / cities. Which is still a big issue across the whole USA.
I, for one, would like to see California crack down on battery cage operations, like they have on other livestock abuses. Other states would eventually follow. There's no reason animals should have to suffer to keep the meat industry productive.