Can pigs get brain freeze?
Cranial nerve anatomy is probably pretty similar in pigs and humans, but I don't think the facial nerve (cranial nerve 7) mediates the pain. That nerve is responsible mostly for controlling movement of muscles we use for facial expression (and a little bit for sensory purposes, from taste buds in the tongue). I’m guessing that the sensation of "brain freeze" comes from temperature receptors we have in our gums, mucosa, and teeth (!) via the trigeminal nerve (cranial nerve 5), which is actually the biggest, most honkin' cranial nerve. It’s so big that anatomists break it down into three parts: upper face (ophthalmic), upper jaw (maxillary), and lower jaw (mandibular). But I digress. It's almost assuredly the same deal in piggies. The sense of your brain freezing might come more from its being such an intensely cold and painful feeling that the neurons stimulate other neurons in the nerve track, and that makes it feel like some of the coldness is coming from your upper face. I haven't had brain freeze in a while, though; I think it’s mostly in the front of your head. I don't think it’s an all-around-head feeling, right?
I'm guessing you’ve noticed a pig eating something super cold and not getting brain freeze (or seeming not to), which could be for a variety of reasons. The first thing that jumps to my head is that pain is mainly perceptual (or central, that is, how your brain is interpreting things) and a pig certainly has a different perceptual framework from a human. Also, pigs are a bit less dainty and don't chew things as much as humans. Maybe they're just swallowing the cold stuff faster and bypassing the brain freeze. Third, their teeth (forgive the tooth fixation—dental school is brainwashing me into a tooth-obsessed freakazoid) might be somewhat more insulated (more enamel and dentin) than a human’s such that the cold stimulus is not as intense.
I hope that helps!
Wikipedia is a great jumping-off point, too.