BEIJING—Political controversy isn’t the only problem American companies face in China. They are also up against adversaries like Three Squirrels.

In just seven years the maker of nuts, seeds and fruits has become one of the country’s most popular snack sellers. Its cartoon squirrel characters have become so popular that there’s now a Three Squirrels theme park under construction and an online show with hundreds of millions of viewers. Its sudden popularity is making it difficult for iconic American snack brands like the Oreo...