No. When you cross you get barnyard variety mixed mutts. Nothing wrong with them. Many crosses make a better layer or a better meat bird.
Sometimes you even get a wondefully delightful looking chicken with exoctic marked plumage. Sometimes you get a chick that looks like one or the other parent. Sometimes you get a chick that has characteristics of both. You won't, however, get a rare breed chicken.
Rare breeds are chicken breeds that are not being kept by many people any longer and their numbers are falling in the population.