Can a neck injury cause eye swelling and severe headache?

Head injury: Eye swelling and severe headache sounds more like a head injury. You can certainly have a head injury associated with a neck injury. Head injuries can be caused by rotational forces without direct impact to your skull (like in a neck injury). You definitely need to get checked out by your doc! i am unclear what kind of trauma was involved; i would get over to an emergency room for an evaluation.

Yes: Eye swelling would be a rare result of neck injury. You should go to the er right away.