Oh, I'm sure Elves never had a squirrel "problem". They probably had a deep and abiding respect and love for squirrels, not for any particular use squirrels may have had, but simply as living creatures different from themselves, unique in their own squirrel-ness, an expression of the grand Theme of Ilúvatar, which gave Shape to the World.
And, squirrels probably made pretty good targets for bow practice.
Ummm....how do you think Legolas got to be such a good shot?