There are two major problems with the argument that Squealer gives for why the pigs should get all of the milk and apples.
First, Squealer is setting up a false choice for the other animals. He is saying that the animals can either do as the pigs say (give them the milk and apples) or Farmer Brown will come back and take the farm again. It seems very unlikely that sharing the food more evenly will cause the collapse of the animals' system.
Second, Squealer claims that the pigs are selflessly using the milk and apples only so that they can help the other animals by governing more effectively. Throughout the book, however, we see that the pigs are not ruling selflessly. Very much to the contrary, in fact. They are actually ruling in ways that put their own goals above the needs of the other animals.
In these two ways, the arguments that Squealer makes at the end of Chapter III are false.