If there are only two alternatives, which is more correct,
"They don't use either of them"
"They don't use any of them"?
I am pretty sure than "any" is more correct, but can I use either "either" ?
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The either version would imply two (2) choices, whereas any would imply multiple choices. While two is a multiple — so any could be used — I prefer either when referring to a specific two choices.
The use of any additionally implies no use outside of the offered subset. So if there is a superset of choices beyond the subset then "They don't use any of them" could imply no use of the superset either.
— Do they prefer Ford or Chevy cars?
— They don't use either of them.
This defines that they do not drive Ford or Chevy cars.
— They don't use any of them.
This may be taken to imply that not only do they not drive Ford or Chevy, they do not drive any cars at all.
— They don't use any of those.
This removes any perceived implication of the superset.
Both are correct, but the "any" option sounds odd to a native-English ear.