Ed M. answered 04/03/16
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You may use double quotation marks within a block quote in MLA style as long as the double quotation marks actually appear within the original material that you're giving in the block quote.
The block quote format signals that you're already giving a direct quotation, so there's no need to distinguish as you would need to do with a "run-in" quotation, i.e., a shorter quote that you incorporate directly into your text, between someone else's words which you'd put in double quotation marks and any of the original's words themselves that are in double quotation marks which you would then give in single quotation marks.