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I don't think this is necessarily a mistake. I'm American, and I think "everyone doesn't X" has a clear meaning of "Not everyone does X". It only fails to mean that if you don't take it as a whole, but no one ever says "Everyone doesn't X" to mean "No one does X"; it always means "Not everyone does X". There are lots of phrases in English that do not mean the same thing as the literal meanings of the words would indi…
I agree with you. However, sometimes such phrases can be a barrier to communication across cultures, and this one especially makes it difficult to teach formal logic. People say "You know what I mean" and that's fine when you're talking to someone from your own culture, or even just to a human. Such an audience will interpret what you say, realise it makes no sense, find an alternative, more sensible interpretation,…
As Rider proposed earlier, some idioms are compressions of logical expressions, whereas some idioms are just mistakes that people have adopted. They're not the sole province of "foreign" countries like India, though. You can find these kinds of grammatically incorrect idioms in America too ("I ain't got nothing") and undoubtedly in England as well.
However widespread "I ain't got nothing" may be, though, it is still incorrect English. Children should be taught that it's not correct English. If everyone starts speaking such loose English, we'll lose the ability to communicate clearly with each other.
I think "Everyone doesn't X" falls into the same category of incorrect idioms that may be widespread, but should be stamped out at every opportunity, except in cases where they're used for humour. If someone said, seriously, "I ain't got nothing to hide,", I would regard them with the same discomfort as if they said "Everybody doesn't need to hide stuff". Both are unfortunate manglings that need to be discouraged where possible. HN seems as good a place as any to do so.