I am using a Logitech attack 3 and it works perfectly in Wing Commander Privateer, so I was a little confused that playing in WC3 it doesn't work at all.

The strange thing about it is that the joystick works in the calibration screen, and even while flying ships the buttons on the joystick work. I just cannot steer.

It's possible that the game defaulted to Mouse Control for you, that happened to me at least. I could fire with the Joystick button, but I couldn't move the ship. Try pressing Alt + O while you are in space to bring up a menu where you can change the control scheme to Joystick, that might fix it.