iPhone 6 crash, boots to B/RSoD or White Square

Hello all. As you can see I am a newbie to the forums, but I am longtime iPhone user. I would not post this unless I could really use the help.

So last Wednesday, I was listening to Spotify on my 64gb iPhone 6. I had earphones on. I walk past a building I wanted to take a picture up, so I pull out the camera app from the lock-screen. I hear the music skip a beat, but it's done that before, I snap the shot and bam, blue screen of death and iPhone crashes. Admittedly this has never happened to me before. Sometimes while just having the phone on the home screen it would randomly reset, but I heard this happens sometimes. I try restarting the iPhone to no avail. Simply white apple logo into a boot loop. I run to my computer to try and update the iPhone, and nope, same boot loop.

I head to the apple store (since I was traveling that night to México), see if they could help me at all. Apparently they could either help restore it or just give me a replacement. I was holding out on restoring the iPhone since I had 2 weeks of pictures I didn't get to back up. That aside, I spend the night trying to troubleshoot the problem before my flight. Many attempts to itunes update and no dice. Finally I give up trying to salvage my pictures and go ahead with a restore... no luck. Attempting restores have given me many weird results. Either the iPhone freezes at the white apple screen. Flashes the blue or red screen of death and loops again. Sometimes its the white apple screen with screeched colors across it. Last time was the apple screen with a perfectly white square at the bottom. I've looked into this white square and have found nothing. I will upload pictures of those two scenarios.

I come to all of you for any guidance or help. Now the phone turns on to the Connect to iTunes screen. I get the errors 4005, 4013, or 4014 from iTunes. I've given up on salvaging my pictures, now I'd just like the phone to work. Yes I could just go to apple and get a new phone, but I'm in México now for the next few months with no apple store for a couple of hours. I've used both PC and Mac computers.

I'm on the T-mobile network if that helps at all. I've read there is a hardware issue with activating wi-fi calling and iPhone 6s.

Any and all help is much appreciated.