Same here

Firefox has been refusing to connect on Google, Yahoo, Youtube, Facebook, etc

Do a malware check with some malware scanning programs on the Windows computer.
Please scan with all programs because each program detects different malware.
All these programs have free versions.

Make sure that you update each program to get the latest version of their databases before doing a scan.

• Malwarebytes' Anti-Malware:
  http://www.malwarebytes.org/mbam.php
• AdwCleaner:
  http://www.bleepingcomputer.com/download/adwcleaner/
  http://www.softpedia.com/get/Antivirus/Removal-Tools/AdwCleaner.shtml
• SuperAntispyware:
  http://www.superantispyware.com/
• Microsoft Safety Scanner:
  http://www.microsoft.com/security/scanner/en-us/default.aspx
• Windows Defender: Home Page:
  http://www.microsoft.com/windows/products/winfamily/defender/default.mspx
• Spybot Search & Destroy:
  http://www.safer-networking.org/en/index.html
• Kasperky Free Security Scan:
  http://www.kaspersky.com/security-scan

You can also do a check for a rootkit infection with TDSSKiller.

• Anti-rootkit utility TDSSKiller:
  http://support.kaspersky.com/5350?el=88446

See also:

• "Spyware on Windows": http://kb.mozillazine.org/Popups_not_blocked

This began immediately after a Firefox update (27 to 27.0.1) and does not occur with any other browser

Upgraded from 27.0.1 when the problem started. I can use other browsers with no problems, though I would MUCH prefer FF. Freeze often occurs when a second (or third) tab is open. However, I can access the http://support.mozilla.org site without problems.

That is, upgraded TO 27.0.1

Google is the main site Firefox is hanging up on most often on my end

No problems with IE, Chrome, Safari

Yep

24/7 Firefox 27.0.1 hangs up on Google