The most important quotes from Boxer are his two favorite sayings: "I will work harder" and "Napoleon is always right." I will include some other important quotations by and about Boxer.
Boxer was an enormous beast, nearly eighteen hands high, and as strong as any two ordinary horses put together. A white stripe down his nose gave him a somewhat stupid appearance, and in fact he was not of first-rate intelligence, but he was universally respected for his steadiness of character and tremendous powers of work. (ch 1)
This quotation is important because it describes Boxer’s most important traits: his size, his strength, his good standing with the other animals and his lack of intelligence. All of these traits make Boxer easy to exploit.
"You, Boxer, the very day that those great muscles of yours lose their power, Jones will sell you to the knacker, who will cut your throat and boil you down for the foxhounds." (ch 1)
Old Major uses Boxer as an example during his speech about how men exploit the animals. This is important because of its foreshadowing and irony. Boxer is indeed sold by Napoleon for his meat and bones once he can no longer work.
Their most faithful disciples were the two cart-horses, Boxer and Clover. These two had great difficulty in thinking anything out for themselves, but having once accepted the pigs as their teachers, they absorbed everything that they were told, and passed it on to the other animals by simple arguments. (ch 2)
Boxer’s loyalty is key to his importance to the rebellion, but his ability to convince the other animals is also key.
When Snowball says that all animals should go naked, Boxer is one of the first to comply.
When Boxer heard this he fetched the small straw hat which he wore in summer to keep the flies out of his ears, and flung it on to the fire with the rest. (ch 2)
This is another example of Boxer's immediate obedience and unquestioning loyalty.
[The] pigs with their cleverness and Boxer with his tremendous muscles always pulled them through. Boxer was the admiration of everybody. He had been a hard worker even in Jones's time, but now he seemed more like three horses than one; there were days when the entire work of the farm seemed to rest on his mighty shoulders. (ch 3)
The pigs depend on Boxer not just for his popularity, but his strength. Since he is well-respected, he sets an example to be followed.