I am installing my new spa and it is set up as a 3 wire 240 VAC, 60Hz unit. My previous spa used 4 wires (red, black, white, and ground). The new one appears to use red, black, and ground. Can I use my old GFCI (made to use red, Black, White) on the new module and have it work properly without using the white/ neutral? Do I need a white/ neutral since it runs on 240 VAC?

Yes the GFI breaker will work properly without a Load Neutral connected to the breaker, Just be sure to connect the white pigtail from the breaker to a neutral bar.