Switching of USB data communication is doable with relays, but not with the selected G3MB SSR type. The G3MB SSR does not specify any parasitic capacitance of output contacts, and list minimum 100 mA as operating current.
For USB signals to work "great" at HS rates, the pin parasitic capacitance must be under 5pF, and ON resistance of the signal switch must be less than 2-3 Ohms.
There are relays that work for USB signaling, including SuperSpeed rates (5Gbps), like Coto 9800 series or Meder CFR05 or Panasonic ARJ or Teledyne S172 types. These relays are designed to provide 50 Ohm controlled impedance along the switched line. To make them work at USB speeds, the PCBs must be designed with controlled differential impedance of 90-100 Ohms, and the screw-in terminals (as shown) will definitely not work for HDD. In addition, this kind of relays are pricey, form $10 to $100+ each.
More, for this switch to really work with any USB, the VBUS also must be turned OFF and ON when switching, so the connected USB device (HDD) can go into non-enumerated state, and be freshly recognized/enumerated by new USB host after the switch. The VBUS switch can be done by any type of relay, since there is no high-speed requirements.
In short, the selected SSR will NOT work great for USB HDD.