Unless that board has a USB encoder on it someplace, I don't think you'll be able to attach it to the USB cable. USB has 4 wires, of which 1 is ground and 1 is +5V power. The other two are transmit and receive from the USB protocals.
If you knew what the pinout was for the wires of the plug (ie, what each one mapped to), if it was a one to one mapping with your buttons, then you could attach a USB encoder to it by wiring it properly, and get USB that way. Considering you have 14 microswitches (buttons + joystick) and that connector only has 9, I don't think you'll be so lucky...