Hello Fabian,
Only the target systems needs to have a public domain name.
In your case the target system is HCI so ERP does not need to have a public domain name.
If you plan to send data from C4C to ERP in that case you will need to use a public domain name like your external.company.com (and an SSL certificate from a CA trusted by SAP).
For security purposes SAP recommends to use a reverseproxy in order to prevent your ERP system to be exposed directly to the internet.
Best regards,
Benjamin