A fix maybe?
I have a CQ62 with the same issue. It's been sitting around for almost a year since the failure since I couldn't find a cheap replacment board. I found motherboards for $60 to 100 on ebay today, but before ordering one I decided to have another go at the machine. I read the forum and was about ready to try windows-key+B at power on, but I think I stumbled onto the fix.
It was shutting down maybe half a second after the red Compaq splash screen came up. Not remembering HP/Compaq CMOS setup keys, I watched a few times and caught "Press the ESC key for Startup Menu", so I started pounding ESC as soon as I hit the power button. (I later discovered that as soon as the splash screen comes up, you can hit pause to stop it, but that was after I stumbled onto the fix).
The Startup Menu came up
F1 System Information
F2 System Diagnostics
F9 Boot Device Options
F10 BIOS Setup (not sure if that one was there before the fix, don't think it was)
F11 System Recovery
ENTER - Continue Setup (This may have been ESC instead of ENTER before the fix).
I ran system diagnostics:
F1 System Information
F2 Start-up Test
F3 Run-In Test
F4 Hard Disk Test
F11 Error Log
ESC Exit
I ran the startup test. It tested memory (100% passed), then the hard drive which also passed (surprising because since I'd given up on this machine, I swapped with its replacement when it started reporting hard drive errors - I had reformatted that drive with the Hitachi DFT utility which did find bad sectors. It seemed to be working at first, but then Windows started taking 15 minutes to boot, looking like the drive was failing again) - I had forgotten I put the failed drive in. The tests completed and I pressed ESC to exit.
It rebooted and I pressed ESC to get into startup menu again. I didn't see anything else that looked useful so I hit enter to continue boot, fully expecting it to shut down. The next thing I knew, I got "Starting Windows"...I expected it to fail at any point, since the image on the drive was from an ASUS laptop but Windows 7 actually booted up using generic video drivers.
I played around in Windows a little, and it worked. I shut it down, expecting that to be the last time I got this board to boot, but when I powered it back on, it booted normally, and has done so about ten times since.
So I guess the answer is to hit ESC as soon as you power on, and run startup test. No idea why this worked but it did. I'm curious if it will work for anyone else. The only difference I see is my CQ62 did not have any BIOS, WIFI or any other errors show up before it died.