Q: Insecure Connection for websites.

Apple update on iCloud.com security - Apple Support

Security Issues Warning List

Contact your school's IT helpdesk.

What is the site?

The site has a valid certificate.

Start up in Recovery mode. In the OS X Utilities screen, select Get Help Online. A clean copy of Safari will launch. No plugins, such as Flash, will be available. While in Recovery, you'll have no access to your saved bookmarks or passwords, so make a note of those before you begin, if they're needed for the test.

Test. After testing, restart as usual and post the results.

If that means you couldn't load the site in Recovery mode, then the cause of the problem is external to the computer. Test on another network, such as a public hotspot or the hotspot created by a mobile phone.

When I loaded the site, I was directed to a login page. That was as far as I could go, because I don't have a login. Can you load that login page?

And it was the same in Recovery mode? You're quite sure about that?