The calculation answering the question in the title is not that difficult, but I needed it recently, and so I thought I would add it to my blog.
Suppose we have a covariance matrix
By definition, the eigenvalues are the solutions to the characteristic equation . From the quadratic formula we obtain
Repeated eigenvalues occur precisely when the discriminant (the expression under the square root sign) is . Thus repeated eigenvalues occur when ,
Thus has repeated eigenvalues precisely when the two random variables are uncorrelated and have the same variance; i.e., and .