Yes, I would say that is a decent example of an unconditioned punisher. It is definitely not an example of a conditioned punisher. An unconditioned punisher is a stimulus of some kind that is naturally punishing to you. Another way of thinking about it is to say that avoiding that stimulus has some kind of natural survival value. Burns cause pain, and your aversion to that pain and getting burned again is a natural aversion. Nobody had to condition you to understand that avoiding burns is a good thing.
Unconditioned punishers can be used for behavior modifications and are common in things like training dogs. For example, if the dog is not walking by your side, a swift leash pull causes a moment of pain. The dog naturally understands that it wants to avoid that stimulus, so it won't wander from your side again. That stimulus avoidance was unconditioned.
A conditioned punisher requires you to be conditioned to the idea that a neutral stimulus is linked to a punishment of some kind. The cookie burn example is not a conditioned punisher because being burned in not a neutral stimulus. The glare that a teacher gives a student is a conditioned punisher. New students don't immediately know that a particular form of eye contact is bad; however, students will become conditioned to understand that a look is often shortly followed by some kind of undesirable punishment. That is a conditioned punisher.