If you follow the tutorial exactly as explained you won’t get any trouble with your website. It will only be “offline” when you change the nameservers. After thats done, there will still be a DNS record cache so people will probably be able to still access your site. As long as you add an new record in your new box for your website! Because you will use your own box for the DNS records, a new record need to be setup there as explained in the tutorial:
If you do have an active website at your domain, make sure to set up the appropriate additional DNS records on your Mail-in-a-Box server. Otherwise, your domain won’t resolve to your website. You can copy your existing DNS records to make sure everything works the same.