Menu bar links are absolute not relative
I went to demonstrate how to do a custom domain with relative links so your site works with https (with the nickname) and separately with your custom domain (with http). Except mine didn't.
It turns out the menu, when used with "page I choose", generates absolute links. It should generate relative links, so that your current server and protocol "stick".
If you use "URL" instead and put in a relative link it works fine.
I have not tried it with the option to generate sublinks automatically.
Obviously there is a work-around, but I think there is so much confusion for people over http/https and custom domains, that if you would make this respect their initial choice (i.e. in reference not during creation) it would be a lot simpler for people.
The darn wordpress mentality of absolute links has littered our world; just say "Nothing is absolute, everything is Relative". Einstein will thank you.