Ezoic
No. Your email won't be affected by the name server change.
The Ezoic system keeps the same settings that you already have in your DNS (the system actually checks your settings for errors at this time). The only change it makes is to point www. and non-www. to the Ezoic system, so that we can serve Ezoic versions of your site to your users.
If you find that you are no longer able to receive or send emails after integrating with Ezoic, it's likely that one or more of your DNS records (usually the MX record) hasn't propagated correctly. To check the records we hold against those at your host, please follow the instructions here.