Hi Linc
I am facing the same problem. I followed both the tests (guest login as well as my own) and faced the same problem. Rebooting did not help. I am using OS X Yosemite version 10.10.5 so not very sure if the same procedure could have worked. Later, I tried your advice from the next post where you suggest to look for the OS X Utility screen, but I couldn't locate it. Then I went to the Utility <help< mac help < connect to the internet < go to a website in safari and tried opening the websites I have been unable to open. No luck! The same problem persists. I followed this second step on my own, an absolute guess work with no luck. I am a computer illiterate and very new to Mac. B/w I closed the turned off the filevault and made another guest account as I have activated Find my Iphone. So it wasn't the Guest user account but the newly created one.
Also, I do not think it can be the problem of the router as I tried connecting it to different routers/internet connections. I switched off this router too but to no avail.
To give you a broader picture; I am able to log into gmail, FB, youtube, but not my University website (www.jnu.ac.in) , newyorktimes webpage and many other webpages. I can google search a topic/term but unable to access the links any further. I face exactly the same problem on chrome and mozilla too. I thought of upgrading to El captain thinking it might help, but the link doesn't work citing the same problem ' safari can't connect to the server'. It was then I realised, that I am unable to access the apple discussion forum too.
I faced this problem after installing safari version 9.1 on 20th april and the security update version on 21st april. Other than this, the flash player from the systems preferences got uninstalled and I am unable to re-install it.
I have tried readjusting the date and time as well as clearing the cache and the web history and other such suggestions but nothing has helped yet.
Looking forward to hear from you.