Your informant is correct. The use of "either" is wrong here.
A correct formulation would be "Both work with words".
"Either" usually means "one or the other".
"Both" means "one and the other".
"I have two sons. Both play the piano."
(Note the plural verb).
"You can come on Monday or Wednesday. Either (day) is okay."
(Note the singular verb)
Occasionally, you have a choice between "both" and "either", particularly when used with the word "side".
"There were paintings on both sides of the wall."
(Note the plural)
"There were paintings on either side of the wall."
(Note the singular)
The correct use of "either" in all its forms is often difficult for non-native speakers. Michael Swan has dealt with the issue well in his "Practical English Usage", which I can wholeheartedly recommend to all learners of English.
Phil W. 9 July 2004
Reply from Phil White (Munich - Germany)