Question: When creating MX records for a domain, is it important what they are called?

I want to manage email through a remote service and have added the appropriate destination and priority. The service that is used to manage the domain (cpanel) does not allow the MX record name to be called `@` so I gave it a name like `mx1.mydomain.com`

Is this is a problem when setting up MX records?

Answer: 
The owner name is crucially important as it has to match the right-hand-side of the email address.

Eg `foo@example.com` -> look up `example.com MX`, `foo@bar.example.com` -> look up `bar.example.com MX`.

If the `MX` exists at `mx1.example.com` it will only serve any purpose for mail sent to addresses ending with `@mx1.example.com` (which I'm guessing is not the intention to have).

As for `@`, this is a special value in the "master file format" (standardized plain text representation of DNS data) which is a shortcut for "current origin".  

What you need to focus on is that you want to add an `MX` at `example.com.` (presumably), not how one might achieve that in some different system than what you are dealing with. If you are working with a GUI, it will come down to how that interface was designed to work. If it's not sufficiently intuitive that you can work out how to do it, you would have to consult the documentation for this specific interface.
You can also give the name for MX record as *domainname.com*. (the period character at the end is required), replace with your primary domain hosted in CPanel.