Little Red Hen is not useful only as a story for little children. It can be helpful in the class of a bit older pupils who are already capable of basic literary analysis s well. This story clearly doesn't qualify as a fairy tale. Apart from talking animals (which are actually representations of two diametrically opposite forces), there are no magical elements. What is even more important, there is no transformation of the main character, the most important element of all classic fairy tales. The conflict is simple, predictable, without escalation, the resolution doesn't change anything.
But there is still a strong moral message, what makes The Red Hen pretty obvious example of a fable (check the article on fables). If we want to find a fairy tale, similar to Little Red Hen, we wouldn't be very successful, but among fables, there is an indisputable favorite: The fable about the Cricket (Grasshopper) and the Ant.
Here is a short condensed info on the difference between fairy tales and fables.
The idea for the classroom: Compare both stories (fables) and have a debate about the characters!