Can someone with a broken neck still? Eat food?

Yes. A broken neck is really a fracture of one or vertebra of the cervical spine. Once it is stabilized by surgery or allowed to heal on its on, if there is no damage to the esophagus or the controlling nerves of deglutition the person will be able to eat any food of choice.
Yes. Whether a brace, Halo or surgery is used to treat a cervical fracture, patients can usually eat.