Kath,
I had a similar problem yesterday and I wanted to share with you my experience.
I have an AT&T 64 GB iPhone 6 which I purchased the day they came out from a local store. I've had no issues with it so far. Saturday night I went out, had it with me the whole night. I didn't plug it in on Saturday night, so Sunday morning when I woke up it was dead. I proceeded to plug it into my computer and it booted up. When the home screen came on, the entire screen was flickering, almost felt like I was tripping or something. The screen and buttons were working fine. When I tried to sync it with iTunes, it would not. Then I got the dreaded blue screen of death, followed by my phone rebooting.
Over the course of the afternoon I saw several things. Blue screen, red screen, red screen fading with blue stripes, blue screen with faded stripes. Each time it would reboot to the home screen, I was able to unlock the phone and use it, but moments later, the whole cycle would start again. I made several attempts to back it up because I was fearful it was on the brink. It wouldn't back up. The phone wouldn't stay on long enough to do that. Eventually, after hours of rebooting, it finally just died. Wouldn't turn on, wouldn't hard reset. Nothing.
I took it to my local AT&T store, as we don't have an Apple store in town. They said it was a hardware issue and that I had to call Apple. I spoke with a lady on the phone, explained to her what was going on and she put up the white flag and said I had a dead unit. All of the forums I've read of this happening with previous models, you could do a hard reset and it would fix it. Others said to just do a restore and again, fixed. Unfortunately, mine was unresponsive. We set up a way for me to send mine in and get a new one or for mine to be fixed.
Fast forward a few hours . . . I'm at work, and all the sudden I hear my texting noises going crazy. WHAT?!?! I looked,and sure enough, my phone had powered back on. Now my screen was still fuzzy/flickering/glowing strangely. Trippy. BUT, it stayed on, and I was able to use it, even upload all of my pics to Dropbox in the case of an emergency (I hadn't backed up my phone since I got the new one). As the night went on, the fuzzy/flickering went away, and today, it's fine. I went home last night and was able to plug it in, no problem, and did a full backup to my computer. I did have an issue with doing that but found some forums on here very helpful. It was saying my phone wasn't connecting, but, come to find out it was because I had a corrupted backup from earlier in the day when I was attempting to back it up. Once I deleted that backup it went through no problem.
I called Apple this morning to get their opinion. They said it was up to me. They still gave me the option of sending it in since it was well under warranty, but, they also said that the Apple Care with the phone lasts for the first year, so, if I'm comfortable, just continue to use it. They of course kept my case on file in case it happens again.
Everything I've read for people having this issue makes it sound like it's a software glitch. I haven't read too much on people having this happen with the 6 yet, even the two people I talked to at Apple said that I was their first caller with this problem with the 6. My recommendation would be to just call Apple. They're going to walk you through all the possible steps to see if your phone will recover. If not, they'll give you the option on how to get it fixed. You shouldn't have to have something brand new act like this. Best of luck!