The Jack characters are a one-parameter deformation of the characters of the symmetric group; a deformation given by the coefficients in the expansion of the Jack polynomials in the basis of power-sum symmetric functions. For each integer partition of n, we give a simple combinatorial formula for the sum of the corresponding Jack character evaluated over all integer partitions of n with no singleton parts. As a corollary, we obtain closed forms for the eigenvalues of the so-called permutation and perfect matching derangement graphs, resolving an open question in algebraic graph theory. Our proofs center around a Jack analogue of a hook product related to Cayley's Omega-process in classical invariant theory, which we call the principal lower hook product.