Custom domain page with SSL
Public status page with custom domain and free SSL certificate (using Let's Encrypt).
Yeah this would be awesome! If this isn't an option, a wildcard cert for status.updown.io (so they could be served securely) would be great. Then you can just chuck Cloudflare on top.
Tamás Márton commented
I've just set up HSTS (HTTP Strict Transport Security) on my site with subdomains option, so none of the subdomains are available via http anymore (only via https).
To make the public status page work, I guess the xxxx.status.updown.io subdomain should serve certificate of the referring status.mysite.com. Probably the cert can be only issued when the CNAME record is set already. Definitely challenging, but would be a cool feature.
Until then, I simply redirect status.mysite.com to the updown.io/xxxx.