Let L(λ) be a self-adjoint quadratic operator polynomial on a Hilbert space with numerical range W(L). The main concern of this paper is with properties of eigenvalues on ∂W(L). The investigation requires a careful discussion of repeated eigenvectors of more general operator polynomials. It is shown that, in the self-adjoint quadratic case, non-real eigenvalues on ∂W(L) are semisimple and (in a sense to be defined) they are normal. Also, for any eigenvalue at a point on ∂W(L) where an external cone property is satisfied, the partial multiplicities cannot exceed two.