Earlier quoted context omitted.
Shakespeare most certainly did speak, and write, Modern English. See http://en.wikipedia.org/wiki/Modern_English . If your claim is that he did not speak an identical dialect to you, well, neither do many English speakers who use double negatives for emphasis (or because they are grammatically required) every day.
This is ludicrous. The language spoken by Shakespeare was undoubtedly and clearly different from anything regularly spoken today, and nearly unintelligible to any English speaker without an interest in Shakespeare or older forms of English. If that is not a different language, then what objective basis do you have for saying that AAVE is a different language?
As noted elsewhere in these comments, the distinction between a language and a dialect is primarily a political one (and some linguists reject it entirely). However, there is a clear line between between Middle English and Modern English which does not exist between Elizabethan and American Standard, and that's why I say the original claim is false.
If what you want to say is, "Okay, Shakespeare used double negatives for emphasis, but he spoke a different dialect than I do, and in my dialect double negatives reverse meaning", that's a fine and fairly uncontroversial statement, in that it forces one to acknowledge that one speaks a dialect. If on the other hand you're trying to say, "Okay, but that was an old kind of English and English today is different," that's factually false.