Does the Hurricane turret gunner have an ejection seat?

It technically has the ejection handle on the seat. I actually have to recheck for how we deal with people in manned turrets quickly evacuating from a ship. So it has the geo and the interaction to support that. It's just, do we actually blast the canopies, which I do not know.

Or maybe shoot it out down below?

You wouldn't want that. So the actual housing that you go into the turret for, it's not a full cylindrical tube. It's actually more rectangular, fit to the seat. And so if we just pushed you straight down, regardless of which way you were facing, there's a chance that you will end up eating your kneecaps down to your ankles and die just as painfully.