Devices no longer able to access certain websites (usually involving https)

Question

Hello,
Starting around 21 Feb a bunch of sites started working really funky.  On Netflix I cannot login.  On flickr I can't post to group forums and other little functionality is broken.  I can't sign into Facebook.  I can't sign into Verizon.com sites (more on how I'm writing this in a second)  The rest of the non-interactive internet works perfectly fine so it's not a DNS issue.
So, there is one computer on which everything is working fine (that's how I got here to post) - otherwise I'd just think it was a network issue.  All the rest of my four computers do not work.  Also, my BluRay player and Roku box no longer can connect to Netflix.  
Since one computer works fine you might be thinking - "oh, he changed something on those others".  Actually, nope.  I even tested on a computer I hadn't used since December and that was working back then, but now cannot access those same sites.  I even booted into a live CD of Fedora Linux - so it's completely clean, DHCP - can't log into Netflix.  Also, I've tried both Firefox and Chrome - both of which work on the computer that is OK.
I ran wireshark on one of the affected computers - basically it's dropping packets on TLS sites.  Just over and over and over again - all I see is TCP Retransmission.  Obviously can't run wireshark on my BluRay player or Roku
First I checked all my settings, rebooted the computers, etc.  Eventually, I called tech support.  Well, first I chatted with tech support.  He had me run ipconfig /flushdns and he had me reset my modem - including pushing the little button on the back with a pen.  Confirmed that everything reset because I lost all my wireless settings.  After the modem reboot I lost him (as I expected would happen, despite him telling me it wouldn't).  Then I got on phone tech support.  Again, they had me check DNS and crap like that.  I know it can't be that stuff because how would I even do ipconfig /flushdns on my bluray player?
On my own, I've tried using Google's DNS servers at 8.8.8.8 as well as using the two nameservers I see in my router's webpage.  (The one at 192.168.1.1)
So what could it possibly be?  A broken router?  I assume if something'd changed at the neighbrhood level, they'd be getting tech suport calls up the wazoo, so it's probably just me?  I'm really annoyed because of the inconsistency - why does one computer work and none of my other devices?  The PS3 was on the fritz around the same time (kept complaining about DNS issues), but it "got better".  
I doubt it's something I've configured on my router since I rebooted it and cleared the settings.  The working computer is on static IP address, but so is every other device here.  Also, the liveCD of Linux was DHCP.  I know people like to come on forums and swear down and up that they haven't changed anything and then it turns out they did a system upgrade or changed something they thought was insignificant - but I literally have touched nothing.  I didn't even realize at first because the wife is the heavy user of the Roku and BluRay player.  And she knows well enough not to touch anything because she hates interruption in service.  (And I double-checked the settings anyway) ANYWAY - the 2 months untouched computer and the liveCD Linux are proof that it's nothing that's changed.  
Anyway, it's been a week now since everything broke.  And I when I got off the phone with tech support they said they put in some kind of ticket for me.  I can't find that on the Verizon website to check the status.  
So I was hoping I could get this problem solved somehow by posting on here.  The guys you get when you call tech support are clearly following a script and have no clue what's going on.  I've loved FiOS ever since I moved into my new house (couldn't get it in my apartment) and I really hate Comcast, but as of right now, my Internet connection is pretty useless.  I can't use my Netflix streaming devices (save the one PS3), and the reason we have more than one computer is so that we can do stuff no matter what computer we're on.  So if I want to manage my Netflix queue, I shouldn't have to go to the one computer in the house that can get to Netflix.
Here's hoping for some help!  Thanks!
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Answer 1

very wierd scenario.  just a few things,  might not be what you're looking for in the way of an answer, but if you could try it on one of the affected machines, and report back to homebase, it could help give us all a good starting point. 
In IE, go to Tools, Internet Options, then the Advanced tab, then make sure "Use SSL 2.0" and "Use SSL 3.0" are checked under Security.
Also try compatibility mode.  In IE, go to Tools, compatibility view settings. 
and as odd as it sounds, it could be a corrupt windows firewall.  try this, it's a tool to repair WMI & The Windows Firewall. http://www.pcwintech.com/shanes-toolbox)  (last download on the page)
I know the blu ray and the roku player throw a monkey wrench into the above, so I am not addressing those yet.  we can and will but I am focusing on the pc's first just to get on good ground,