Is there any way to get rid of the default email account that is created for each cPanel account? In the end, I'd just like to have *all* email go into a new pop account (mailqueue@domain.com) that another mail server of mine can then work with.

No it is part of Cpanel account setup. Also it does not always work so never advise anyone to use it, if you already have a user setup and they want to use that email address then it is best to recreate it in cpanel.