The "unsecured" bit I expect is just to gain access to the internet link on your router. No password needed. Though doesn't a user have to have some login username and password given them anyway, in order to log into the FON service?
"Secured" websites (https

are achieved by the setup at the target company's website. I would expect those to function as normal.
Try it! Connect something to your own FON and go to the login page for your bank, or similar. No need to enter anything, so no danger. You should see the URL with https, and probably a lock symbol however your browser displays that.
My broadband basic info/help site - www.robertos.me.uk
My domains,website and mail hosting - Tsohost.
Internet connection - Plusnet Value Fibre.
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