What was particular about the way Squealer spoke to the animals in Animal Farm?

To understand Squealer's particularity in speech, one should understand his character and also his role and position within the context of the novel.
Firstly, as a pig, Squealer was in a privileged position after the Rebellion - he, just as much as the others of his ilk, took advantage when they were put in leadership positions. Obviously, once they had become accustomed to their new lifestyle, they had to protect it. In this, Squealer became the ideal tool to manipulate, mislead and deceive the other animals in order to propagate and defend their position. It is clear that he was ideally suited for this purpose as he had the skill and persuasive charm to persuade the other animals.
Orwell describes Squealer as follows:

The best known among them was a small fat pig named Squealer, with very round cheeks, twinkling eyes, nimble movements, and a shrill voice. He was a brilliant talker, and when he was arguing some difficult point he had a way of skipping from side to side and whisking his tail which was somehow very persuasive. The others said of Squealer that he could turn black into white.

It is evident from the above that Squealer thoroughly enjoyed his duty as propagandist. The fact that he skipped from side to side whilst whisking his tail made him seem much more persuasive. It is clear that these actions do not only indicate his pleasure but are also used as a distraction. One should understand that, since the general animal populace on the farm are not very intelligent, they cannot be assumed to be the most attentive of listeners and would easily be drawn to Squealer's clearly visible actions than to what he is actually saying.
Added to that, Squealer purposely uses somewhat high-handed and especially emotive language when addressing the animals. The effect is that the animals would not question what he says since it sounds intelligent and reasonable. Evidence of this is found in his speech about the disappearing milk and the windfall apples:

“Comrades!” he cried. “You do not imagine, I hope, that we pigs are doing this in a spirit of selfishness and privilege? Many of us actually dislike milk and apples. I dislike them myself. Our sole object in taking these things is to preserve our health. Milk and apples (this has been proved by Science, comrades) contain substances absolutely necessary to the well-being of a pig. We pigs are brainworkers. The whole management and organisation of this farm depend on us. Day and night we are watching over your welfare. It is for YOUR sake that we drink that milk and eat those apples. Do you know what would happen if we pigs failed in our duty? Jones would come back! Yes, Jones would come back! Surely, comrades,” cried Squealer almost pleadingly, skipping from side to side and whisking his tail, “surely there is no one among you who wants to see Jones come back?”

Squealer uses specific persuasive language techniques: so-called factual evidence (Science); empathy (for YOUR sake); rhetorical questions - (You do not imagine...? Surely..? Do you know...?).
To add impact, Squealer adopts an almost pleading tone and uses his signature charming technique of 'skipping from side to side and whisking his tail.' With such a clever orator addressing them, the poor animals don't stand a chance. Squealer's voice goes practically unchallenged. He is only doubted and questioned by Boxer twice in the entire novel, but that is a different story.
 

Animal Farm is intended to be an allegory of the Soviet Union (U.S.S.R), the now-defunct superpower that, at its zenith, was the world's largest "Communist" state. I use quotation marks around Communism because, in all honesty, the Communism that has taken hold in governments bears little semblance to the idea that Karl Marx put forth in The Communist Manifesto. Communism was characterized by a tight control on information that was disseminated; should something not adhere to or follow the party line, then it was quashed.
The way Squealer spoke to the animals was meant to evoke the imagery of the Soviet-controlled media, the newspaper Pravda in particular. The news Pravda reported had a very strongly pro-government slant overall, skewing basic facts in support of the government and outright distorting if there was no way to make the facts work. Squealer's aim was to twist everything that the pigs did to make it seem like they were adhering to "Animalism", the Communism-like manifesto that drove the actions of the animals in overthrowing the previous tenants of Manor Farm, and also to rewrite the seven core axioms in order to further support their position.
Truth was, in effect, what the government said it was. The acquiescence of the animals to this reflects the state of the U.S.S.R. back when the book would have been written (immediately after World War II), when questioning the government line could easily have been fatal.