People normally say: Shouldn't you do something.... However it would sound strange to say: Should not you do something. The problem is that English contractions like shouldn'tare not the same grammatically as their full versions. Shouldn't you eat less meat? is grammatically correct, by common usage. However, Should not you eat less meat? is not grammatically correct in Modern English.
When people ask questions, the contractions shouldn't and don't are used as if they were single words. In some ways in English they are a single word. Instead of being split, they were moved together to the start of the sentence:
• You shouldn't eat meat → Shouldn't you eat meat?
• You don't eat meat → Don't you eat meat?
The word shouldn't is part of a group of related grammar words. (It is called a modal auxiliary verb negation.) Other words in this group are: can't, couldn't, won't, wouldn't, mightn't, mustn't and oughtn't. The same structural problem can happen with these words.
Originally, shouldn't was a replacement of shouldnot. But in this type of situation it is no longer a replacement of should not: and no longer an opposite of should. Unlike should not, shouldn't is now a single word, and it behaves as a single word, moving as a single word to the start of a question asking (interrogative) sentence. In many questions shouldn't can be replaced with should and keep the same meaning of the sentence. Treating shouldn't as a single word is, in fact, the grammatically correct thing to do in Modern English.