Just got back from the Apple Genius Bar.
In December, my iPhone 5S (which was just over a year old) started showing the blue screen and restarted. It got more and more frequent over time. Since I was abroad, I decided to wait until I get back to the US in January to deal with it. I tried restoring the phone from fresh in iTunes. Once I did that, the phone showed the "Connect to iTunes" screen and wouldn't restarted. Trying to restore (not from backup) did not help.
Took the phone to the Apple store. They looked at it and said there is a graphics card issue and the phone needs to be replaced. Since the phone was (JUST) over a year old, I was out of warranty and needed to pay out of pocket to replace the phone. Great....
A week later I ordered an iPhone 5S for work (through my employer). Guess what? I am now seeing the same issue! At this point, my work phone is in infinite loop of restarts.
So, let's sum up:
Personal phone: bought in Dec 2014, black, 32GB, AT&T.
Work phone: bought in Jan 2015, white, 16GB, Verizon.
Clearly, different phones, yet showing the same symptoms.
Back to my trip the Apple store today: "oh, clearly a hardware issue! Your phone needs to be replaced." Me: "clearly the same issue as my personal phone. Seems like there's something defective in manufacturing." 'Genius' employee: "yes, it looks like the same issue, but your personal phone is out of warranty."
So there you have it! Instead of RECALLING the phones and fixing them, it's just easier to replace those under warranty with refurbished ones (or whatever marketing name Apple gives it), and have customers pay for replacements.