There are four factors to determining Fair Use of copyrighted material that must be weighed in the balance:
Purpose and character of use...better if not for profit; better if transforming in some way
The nature of the work...factual is better, creative works less so
The amount and substance of the work...less is better; uses that don't get at "the heart of the work" are better. There are no rules about percentages, etc, despite myths to the contrary.
The effect of the use on market value...if it doesn't take away from real or potential profit, better
Depending on the nature of the book you are writing, you are probably quite safe using screenshots. You are likely writing something that will stimulate sales and that isn't going to reveal workings of the game that are protected.
Your publisher should have experience with this and probably has a policy for when they ask permission. Unfortunately, many people and organizations don't understand Fair Use and err too far on the side of caution, ceding rights they don't actually have to.
I teach about Fair Use regularly; you can find more information here: http://iteachu.uaf.edu/develop-courses/constructing-a-course/copyright/