I have the same issue as jwebb88.
iCloud on my ASUS ZENBOOK [Model UX303L (64 bit) running Windows 10 Home, Version 1511] cannot log in. It always returns a login error message: THERE WAS A PROBLEM CONNECTING WITH THE APPLE ID SERVER
I followed the kind guidance from Jason L (link to "If you can't connect to the iTunes store") - but I didn't find any answers down this path. Like jwebb88, I too have the latest iTunes installed, and it logs into the Apple store just fine. This strikes me as odd, since I presume both iCloud and iTunes are going to the same place to validate logon credentials.
Key points:
1) I remain logged into my iCloud account just fine via multiple iPhones, multiple iPads, another ZBOX PC running the same version of Windows 10. Confirms that this is not a "User" problem with me trying to use the wrong password.
2) From my problematic laptop, I can log into the internet version of iCloud just fine - via both Chrome and Firefox.
3) As already noted, from my problematic laptop, iTunes launches just fine, and logs into my iTunes store (...in Australia)
4) I've made sure Windows is fully up-to-date, and that my Office 2013 is also fully up-to-date, with complete shutdown/restart of the laptop to ensure nothings half-installed.
5) Before upgrade of iCloud to the latest version a couple of weeks ago (I'm now on Version 5.2.1.69), iCloud worked beautifully, including proper sync of Outlook calendar and contacts, and bookmarks across different browsers. I'd love to try and reinstall the older version - see if that rolls me back to my earlier nirvana, but I haven't been able to find anyway to do this.
Any suggestions as to best "next steps" would be warmly appreciated...