Turn-off voltage for SiC modules, does it always have to be negative?
Depends.

You need to ask yourself why a negative voltage is recommended. If you look at the datasheet for the specific part in question, the gate-source threshold minimum is 1.6V, this means it is guaranteed to be OFF for voltage below 1.6V. Likewise it is guaranteed to be active for a voltage above 4V.

presenting the gate-source with 0V would be below the 1.6V threshold to guarantee the device is OFF, in a non-dynamic situation but this might not be always the case. Take a H-bridge or an inverter, there are other switching devices all producing their own dv/dt. Now consider the parasitic capacitance of the MOSFET. How do you ensure the device is held-off when you command it off, in a power circuit with other switching edges. With a risetime of 70ns, it does not take much capacitance to cause a significant current to flow and potentially turn the device back on.

Driving a MOSFET with Negative Off- State Gate Voltage