Jeeze, you've all got it so
wrong.
Eeyore is depressed. Yes. Duh. He's not ashamed of his body. He's not like the rat from Charlotte's Web (Templeton). He's not having some kind of existential crisis brought on by the realization that he's just the figment of Christopher Robin's addled little mind.
Eeyore is a device that the author of the stories used. Sure, if Eeyore was real he might start having some sort of existential crisis, but HE'S NOT REAL. He's a character created by the author for a purpose.
What is that purpose? It's not to say that being depressed is alright. It's not to point out weird Kantian quirks in the realities of children's fiction. It's not to provide a bouncing post for Tigger. Eeyore points out all those things in the world that are good.
He almost always is depressed. He finds the down side of everything. And, just when you'd lost hope, Eeyore finds a butterfly, or a tree blowing in the wind, something of simple, pure beauty. The contrast is what he's for. Pooh just wanders around on a perpetual acid trip. Tigger is also on drugs, probably speed. Piglet needs some drugs, badly. And Rabbit... well, Rabbit is much more of a downer than our favorite mule. So, Eeyore is the only one who looks at the world with at all accurate eyes. And he finds beauty in it, sometimes. That sometimes, that' what keeps us looking, searching. It's what keeps us going.