Eeyore is always sad and yes I do believe he would be considered clinically depressed if he were a real animal or person.
But after all he's just a fictional character but he does seem depressed and I think they made him depressed to teach children how to deal with real life situations and sadness even though some children or most children cannot understand exactly what depression is.
He does seem always depressed though and sometimes I can relate to Eeyore's depression as I sometimes feel just like him but after awhile the depression goes away unlike Eeyore's depression.
I would say that if Eeyore were a real animal and got diagnosed with depression he would be clinically depressed.