plug the extrernal drive into a different USB port and see if that fixes it. If your quickbooks file is on the external HDD then it will not work until your HDD works.
Get into Device manager and see if the USB drive is listed. Get into BIOS and see if the drive is listed (should show up under USB options, or the boot order menu, it will not be listed with the rest of the drives).
Try the drive in another PC and see if it will recognize it. WARNING: many viruses will attach themselves to USB drives, so make sure that any PC you hook it into is protected, and that autoplay is turned off. Plugging it into a Mac or Linux box will prevent this from being an issue.
If it is not recognized by another PC then it is time to remove the drive from the housing and then either install it internally if you have space (free), or put it in a new housing ($25-50).