Note that if you let your domain lapse and someone else registers it afterwards, that person will also gain control over your e-mail address (and likely all accounts associated with it, if they are not secured with an additional factor of authentication / recovery).
As long as you're very thorough about removing any linked connection from your expired e-mail, you should be OK. That includes all accounts that you registered using this e-mail, as well as all e-mail contacts that you've built up using that account.
I think the person was saying it's a good idea to have your own domain because with a gmail.com address, you're stuck with Gmail. With your own, you can change providers any time by setting up your addresses at the new provider and updating the mail records to point to them. Boom! New email provider, same addresses.