Yes, “none of them” and “not any of them” are equivalent in terms of meaning. GETAG, however, suggests using the phrase “none of them” in your writing. “None of them” is more concise and improves clarity. In today’s busy world, there is an advantage to being able to express yourself using as few words as possible.
Examples in published writing
Here are two examples of modern usage of “none of them” in writing:
by Jeremy W. Peters, New York Times
September 4, 2013
“When he visited a grocery store on Wednesday, he said, almost a dozen people told him they thought intervening in Syria was a bad idea. None of them expressed support.”
by James Downie, Washington Post
January 15, 2015
“But now we learn that the agency’s internal panel has decided none of them should face discipline.”
Replacing the phrase “none of them” with the equivalent phrase “not any of them” would only make these sentences awkward. Therefore, avoid “not any of them” and use “none of them” whenever possible.
Whether or not any of them
The phrase “whether or not any of them” is commonly used in English. The expression is the combination of the two phrases, “whether or not” and “any of them“. The following example is from the New York Times.
by Ian Austen, NY Times
October 29, 2010
By some accounts, 125,000 medical miracles have been attributed to Brother André. But Donald L. Boisvert, a professor of religion at Concordia University in Montreal, said that whether or not any of them actually occurred was almost beside the point.
Word Count: 285