The previous post was quite thorough. I would only add that the fundamental reason why the animals believe with Squealer is because of his ability to "spin." Squealer's ability to have a seemingly credible answer for everything that benefits Napoleon's growing power control allows the other animals to believe that what is being said as constituting fundamental truth. Squealer's ability to make it seem that it is both a present condition and historical inevitability that has led to Napoleon's seizure of power also impacts the other animals in order for them to believe that what is being said is right. I am not sure they "agree" with Squealer, for there is no other story out there. The constant fear of the humans returning is something that Squealer uses at any point he feels that the animals' focus is wavering. With this manipulation intact, Squealer is able to convince the other animals of the legitimacy of Napoleon's rules and order, confirming the power of a government controlled media.
When Orwell introduces Squealer in Chapter 2, he describes him as a very persuasive talker who jumps around and wiggles his tail in a way that makes others believe him. He is said to be able to make them believe that "black is white."
In chapter 7, Squealer recreates a questoned event very graphically and, "when Squealer described the scene so graphically, it seemed to the animals that they did remember it."
Another means of persuasion he uses is to convince the other animals that they are stupid and he is intelligent. He does this by using words that they don't understand, like "tactics" (chapter 5)
Also, he begins to use the threat that if they don't believe him, somehow that will cause Mr. Jones to return to the farm--something that none of the animals want. He never explains how a leads to b, he just leaves the threat hanging there.
Then, he gathers some dogs to look mean and growl in chapter 5 so that if the people don't agree right away, they feel threatened.
In chapter 6, when some of the animals begin to object and say that there had been a prohibition against trade, Squealer asks them,
" Are you certain that this is not something that you have dreamed, comrades? Have you any record of such a resolution? Is it written down anywhere?" And since it was certainly true that nothing of the kind existed in writing, the animals were satisfied that they had been mistaken."
In the same chapter, when some of the animals begin to question the rightness of the pigs, who have moved into the farmhouse, sleeping in beds. Clover thinks that she remembers one of the commandments being against beds, but she can't read the list herself. She gets Muriel to read it for her and discoveres that it says "No animal shall sleep in a bed with sheets," This is obviously one of the commandments that has been changed after the fact, but since nobody ever actually sees Squealer changing the commandments, then they have to believe that it is their own memories that are to fault.