With controller holding jump rises and crouch lowers.

Trying to figure this out too. Are there two versions of the Hover Hog? One that simply hovers, and one that flies? Because I've tried holding jump and crouch, and they both seem to do nothing.

I am also having issues with the hoverhog not working. The only one I have come across was the one in Assault on the Control Room, and it doesn't seem to fly.

That one isn't the flying one. Later you'll find one that goes from normal wheels to hovering as you get in, that's the one that can fly.

stop moving and press jump, you cannot do the two at the same time. Also it sometime pop out of a d20 grenade

Flying the DeLorean Hog is affected by camera position due to it moving the vehicle's orientation. This is most relevant to keyboard players. Holding jump on either controller or Keyboard will raise the hog, while crouch will lower it. This is affected by the angle the hog is at. If your hog was tilted 45 degrees backwards, raising it will push it along that 45 degree angle. If your hog has guns attached to the sides, it cannot fly.