I think that Boxer fits into the climax in a couple of ways. The resolution offered is one where the pigs are able to consolidate all of their control over the animals. They have become "the new boss," without demonstrating any real difference from the humans. They wish to control everything, and this includes history and memory. They do not want any remnant of the past and this is why the pigs have never retired any animal, though the promises about that they will. In the final analysis, the conclusion that shows that no animal can stand in the way of what the pigs do or for what they wish.
Boxer fits into this condition in a couple of ways. The first is that he has outlived his usefulness as a worker, and thus Napoleon sees it fit to send him to the Knacker's. When Boxer's lung collapses, his usefulness is over. The pigs would not want to keep Boxer on the farm because he is such a well respect figure amongst the other animals. No animal could ever say anything bad about Boxer and all came to admire his excellent work ethic. Boxer not working is more dangerous to the pigs' control more than anything else, as he could be seen as a "cult figure" or one that would rival the pigs' desire for undivided attention. In this, Boxer becomes part of the pigs' desire to consolidate control when he is sent away. In wishing to severe all links to the past, the pigs end up sending Boxer, one of the oldest animals on the farm, away so that he cannot be seen as anything but a figment of the past that pigs like Squealer can rewrite to their own benefit. In this, Boxer becomes part of the novel's conclusion and its own thematic development.