You CAN crossbreed chickens!
Here's some context. When I first started playing with this new character one of my top priorities was to find/collect some animals to start my own farm. And in the lower part of Muldraugh (where I live now) I found some chickens.
It was about 4 Leghorn (White) hens and 2 Rhode Island Red (Brown) hens with 2 chicks that were both Leghorn and male. As keep playing I would find the odd chicken here or there, but I never found a rooster only hens. This wasn't that much of a problem because I already had the 2 male chicks, so I just had to wait for them grow up to get roosters. The issue was that they were both Leghorn and I wasn't sure if they would breed with the Rhode Island hens.
Finally, I got my answer and yes, they will.
From what I observed the mother hen's offspring will always be the same breed has her no matter what breed the father rooster is. So, if you want breed brown chickens you need a brown hen, and if want to breed white chickens you need a white hen. The rooster's breed doesn't matter.
I believe this applies the same to other animals like sheep, pigs, and cows but I don't have enough testing to be sure.
Wanted to create this post to shed some light on animal breeding.