Problem with iOS 7.1 upgrade on iPhone 5S

I upgraded my phone on March 10th at about 1700 central time. Once done, I noticed that the screen was flickering, so I rebooted the phone. The device did not come back up, it only flashed the white background / black Apple over and over. At the time it was connected to power (the brick and cable that came with it) so I unplugged it to see what would happen. It shut down. I was not able to get it to boot until I plugged it into my computer, and only then did the process of holding down all buttons for a full reset work. Once done, I walked through the process of setting it up and restoring from backup. Everything worked fine, so I was terribly surprised when I woke up at 0800 on the 11th, since I was supposed to wake up at 0700. I rely on my iPhone as my alarm clock and it had died overnight. The battery was not the issue, but whatever happened before had happened again. I called Apple support as I was going back through the process and they agreed that it was the right way to handle it and told me to let them know if there were any more problems (Case #586872875). I had no issues the rest of the day. However, at 0330 this morning I woke up and wanted to check the time and found that the phone had once again done it's thing (I know that it was 0330 because my iPad still worked). So, I did the process again, however, this time there was no eventual restore, When iTunes shows restoring firmware, a reboot takes place that puts the phone right back into a continuous reboot cycle until I disconnect it from the computer at which time it just turns off.

Update - On my 4th try, it actually booted after being disconnected from the computer after the firmware phase was complete. So, instead of restoring it again from the backup, I set it up as a new phone and got as far as configuring my fingerprints before it died again. As I type it is in a continuous black screen / white screen with black Apple cycle. The computer is recognizing it for only a moment before the USB disconnected tone sounds. "iTunes could not connect to this iPhone. An unknown error has occurred." is showing up on iTunes, one error stacked atop another...and another...and another.

The phone has always been in an Otter Box and has never been wet, dropped or in any other way abused. I am an admin on the compter in question, I tried rebooting it, I checked drivers to ensure they were the newest, iTunes is up to date, other USB devices disconnected,

My iPad Air was upgraded at the same time and has been working perfectly since.

Working on try #7. No change.

I'd say that my 7.1 experience has been iffy.

Any assistance would be appreciated.