Currently cPanel will start with eth0 and work through eth1 when you assign nameservers to your IPs via WHM. Unfortunately, this will automatically choose your management IP range first, which of course are non-routable. There are two work-arounds for this:
1: Create a fake nameserver record on your private IPs, and add an A record for that nameserver to your DNS. This is the fastest option, but not the best.
2: SSH to your server and edit /etc/nameserverips manually. An example file is below:
Just replace the ‘0’ after the ‘=’ with your nameserver name. In this example, the first IP (10.x.x.x) is the management IP, and not used for nameservers. The next two IPs are public and assigned to two nameservers. The last three are public and not in use.