How to represent a user who hasn't churned in training data
In your setup, the only way is probably to set it to some large number, say 365. However, this will force you to discard all dates that are less than 365 days old, because you can't be certain a customer won't churn before he reaches 365 days (which is still in the future).

A better and more common way is to look at churn on a rolling window basis, e.g. will the customer churn within the next 30 days. This turns your problem into a classification problem. That way you only need to discard the most recent 30 days of data.