What carrier are you with?
A specific app is probably the cause. The day before the Verizon 4.4.3 update came out I had just closed out of checking my text messages when I got a pop-up about Google+ or Google Now (can't recall which) had stopped responding (which was odd because I never used the app, and it had never been listed in my running list). I tried to click the button options that came up with the error message (can't recall what the two options were), but it was unresponsive. I thought perhaps the error was slowing down the phone, and it would eventually respond, so I left it to sleep. Later I looked at it to discover that the phone wasn't responding (the screen was black, and pressing/holding the power button did nothing), and at one point I could make out that there was a faint Verizon logo (like what would show during a boot). Eventually I did what you guys did with power and volume to reboot. I've never had the problem again after updating to 4.4.3
To make a long story shorter it was caused by an app, and yours likely is too (not necessarily an HTC One so you shouldn't expect HTC to fix an issue that is not caused by their basic pre-installed software. If you can find what app is at fault you can likely stop it from running in the background (disable it). Make sure all your apps are the most up-to-date versions. Try running the phone in safe mode for awhile (which will disable the 3rd party apps), and see if you have the issue. If there are no problems in safe mode boot back into regular mode and Force Stop all but one of your third party apps listed in Running. If everything still works re-enable another one until you find the app were the issue occurs again. Report that app to its creator in hopes that he/she/they may be able to fix it.