I have used a domain on my trustworthy x10hosting account. Pretty simple, go and change the nameservers to Cloudflare and point x10 to the domain via an addon domain.
Then use the flexible ssl proxy as the certificate cannot be installed on the free cpanel (it might have been a thing before to...
Amend your robots.txt to disallow all search engines from seeing a directory on that host.
Set .htaccess permissions on the folder - make it password protected or something!
The fact that they are searchable on Google shows you probably have no htaccess denying access from those files. I don't...