Should you register your domain with the same people that host your website? The answer depends on who you talk to and what their opinion (or previous good/bad experience) is.
Many people prefer to keep control of their domains with a central registration service independent of their hosting providers. This is very useful for those who own or manage a large number of domains, and especially for those who use multiple web hosting services. Another benefit of registering your domain through a third-party is that you might gain a better insight on how the overall components of Domain Name System works to match your domain to your web host.
Some people also advocate that if the web host goes out of business, the domain can simply be pointed to another web host and not be affected (those advocates sometimes fail to mention that Registerfly, an ICANN-accredited Registrar, tied up a large number of domain names during their legal and financial troubles. Domain registrars are not exempt from problems).
Greenix360 partners with a Registrar to provide domain registration services as a convenience for its customers. This partnership requires that Greenix360 provides a certain level of customer service and accessibility to the domain. This includes managing the nameservers, enabling/disabling the registrar lock and retrieving the EPP key/authorization Code to transfer the domain registration to another registrar (which you can do at any time).
Ultimately, if a customer of any web host is having problems regarding a domain they registered through that host, the customer has a level of recourse by contacting the sponsoring Registrar and detailing the nature of the complaint with the hosting company who registered the domain.
To find the Registration info of a domain, you can use the WHOIS search at DNS Stuff. Enter the domain name in the WHOIS search field and you should see the actual registrar, contact info for the domain name, and the host who provided the registration services.
In short, if you register a domain name with Greenix360 there is nothing to worry about. You can contact Support if you need assistance with managing a domain registered through Greenix360, or wish to transfer it.
Who you choose to register your domain name with is your decision and Greenix360 will be happy to provide any information in order for you to make the best decision for your needs.