An HP BIOS is unlikely to work on a non-HP computer, even if it's the same brand and model number, etc.
Get the BIOS from Asus. Pay attention to different revisions of the motherboard, if that applies.
As for Operating Systems, the BIOS doesn't care (or know) what you're running on the computer. The only hitch might be if you were trying to flash the BIOS from within Windows. Then the
program you use to
perform the flash might care, but not the actual BIOS file it is flashing. I would recommend flashing the BIOS from a bootable thumbdrive, if that's allowed by the flashing process.
PS: Your link won't work. List the exact model and revision number of your board.
PPS: You might have got a dodgy board off eBay. I'm not saying you did, so don't go complaining yet. But it's happened. First things first, let's try the other stuff.