The short answer is: no. There are no doctrinal requirements in Zen practice.
The longer answer is: what is there to be reborn? I.e., what is your fundamental nature? what is my fundamental nature? what is every human being’s fundamental nature? That is the basic question, independent of any belief or disbelief in rebirth. Zen practice is a way of perceiving our fundamental nature, fully realizing it, and acting accordingly. This nature is independent of any doctrine, independent even of speech and thinking. There are a lot of words and explanations out there about it. But you have to find it yourself.