Well, that is strange. Sounding more and more like a motherboard chipset has failed. If you have USB2 still working it may not be worth a new motherboard. If still under warranty then contact HP.
If you want to keep this motherboard there is a way to get a HP USB3 card and have 2 ports on the backplane and two up front..... I have posted on how I do this for the Z400/Z600 workstations. That is the newer HP USB 3.0 card, called a "2x2" card for the 4 total ports, and is based on the same TI chipset that is on the Z620 motherboard.
Again, I'm assuming you have done a thorough job of trying to reset everything on the motherboard using all the tricks posted about in this forum.....