Jack is obsessed with killing the pig for reasons of political power, and for personal validation.
In Chapter 1, Jack is essentially given a title and task; hunting. However in the same chapter he is unable to kill a pig, presumably because he can't stomach actually performing the act. He vows not to hesitate again. Thus he needs to kill a pig in order to complete this "journey," or arc, that represents his redemption for failure. This is a fairly common theme in character development.
In broader terms, Jack and his hunters all fail their task; neither Jack nor any of the hunters are able to catch anything. In social and political terms, this means they aren't pulling their weight, and because one's job becomes more unified with one's worth and social standing under the circumstances, this means that they have no value, endangering them and calling their purpose in the society into question. Thus, Jack needs to kill a pig in order to demonstrate that he and the hunters have value and can contribute to the society.