In this work we study the point spectra of selfadjoint Sturm-Liouville operators with generalized point interactions, where the two one-sided limits of the solution data are relatedvia a general SL(2,R)matrix. We are particularly interested in the stability of eigenvalues withrespect to the variation of the parameters of the interaction matrix. As a particular applicationto the case of random generalized point interactions we establish a version of Pastur’s theorem,stating that except for degenerate cases, any given energy is an eigenvalue only with probabilityzero. For this result, independence is importantbut identical distribution is not required, andhence our result extends Pastur’s theoremfrom the ergodic setting to the non-ergodic setting.