As an adjective, prize means “likely to win a prize”. This means that if the boar entered a contest, it would have a good chance of winning. It doesn't necessarily imply that the animal has ever entered a contest or ever will, just that it has the qualities that would make it a likely winner. This is mostly applied to animals (prize cow) and to plants (prize orchid), but can also be applied to other concepts (prize essay, prize performance, …).
Prize-winning would mean that the animal has won at least one prize.
Prized is rather different. A prized possession is something whose owner puts a high price on, i.e. something that the owner considers to be so valuable that he does not wish to part from it. It can either be objectively valuable, in the sense that it would fetch a high price if the owner sold it, or subjectively, in the sense that the owner does not want to part from it, perhaps for sentimental reasons. In the subjective sense, treasured is more often used than prized.