Yes, to use your existing Ring4 account and phone number on your new Phone you just need to logout from Ring4 on your old phone and login to Ring4 with the same account on your new phone.
The history of the messages is no transferable as no copy is kept on the Ring4 servers.
You need to login via the same credentials on the new phone that the ones you use to create the account on the old phone. e.g. If you used a Google or FB account to connect, you need to use the same one.