This Forge 1.20.1 mod adds extra compatibility between the following mods:

Incubation
Vanilla Backport
Mystic's Biomes
Nest Egg

Here's a rundown of what this mod adds:

Egg crates and hay nests for all egg types - warm, cold and rainbow eggs plus twig nests for warm and cold variants.
Warm, cold and rainbow chickens now properly interact with hay nests and can lay their respective eggs
A custom registry to handle interaction of extra warm, cold and temperate datapack variants with nests and egg items (more info below).
Mystic's Biomes rainbow chickens and VB variant chickens now incubate and can hatch from their respective eggs in nests (Nest Egg required)

Twig nests don't generate in the world as that would create a hard dependancy on Incubation. Instead they're simply there to allow modpack/datapack creators to create biome-based twig nest worldgen feature.

Handling custom Chicken variants

Data-driven chicken variants interaction with nests and eggs is automatically handled by biome tag lookup of each variant entry. Here's a simple list showing which tag corresponds to which variant:

#minecraft:spawns_warm_variant_farm_animals - any chicken added to this tag will lay, hatch from and interact with Brown Eggs
#minecraft:spawns_cold_variant_farm_animals - any chicken added to this tag will lay, hatch from and interact with Blue Eggs
any other tag or no tag - any chicken added to any other tag or without such tag will lay, hatch from and interact with Temperate Eggs

Blueprint is required, other mods are optional.

Please report bugs. I have tested the mod in a heavily modded environment without any issues, but your results may vary. I will try to squash them as soon as I can.

Eggcelent (Compat)