It appears that the characters all represent a different mental illness. Eyeore is obvious, but now that I really think about it, it all makes perfect sense. Let's take a look at what is potentially bothering the rest of the residents of the Hundred Acre Wood.
1. Winnie the Pooh
It is pretty clear that Pooh Bear struggles with ADHD, among others, but severe ADHD appears to be the main concern. His scattered thoughts, disorganized lifestyle, randomness and constant forgetfulness all scream ADHD to me.
No matter how many Winnie the Pooh stories I read my kids, or the number of shows I watch with them, I can count the number of time I have seen Eyeore happy on one hand. He is always sad or depressed, and he is likely the saddest character in the history of children's books. It is clear he suffers from severe depression.
The son of Kanga, Roo seems to display symptoms of being on the autism spectrum. Roo seems to operate on two opposite ends; sometimes he doesn't pay attention to anything that is going on around him, and he ends up in somewhat dangerous situations. Other times he decides to sit quietly in his mom's pouch, completely ignoring the world around him. This says potential autism to me.
If there was ever a textbook case for anxiety, Piglet would have his picture below the definition. He is constantly worried about what might happen, and sudden surprises like noises and movements cause him to run and hide. Piglet enjoys things to be calm and simple.