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" ?
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.
This defines that they do not drive Ford or Chevy cars.
This may be taken to imply that not only do they not drive Ford or Chevy, they do not drive any cars at all.
This removes any perceived implication of the superset.