Given a compact strictly pseudoconvex CR manifold M, we study the differentiability of the eigenvalues of the sub-Laplacian Δb,θ associated with a compatible contact form (i.e. a pseudo-Hermitian structure) θ on M, under conformal deformations of θ. As a first application, we show that the property of having only simple eigenvalues is generic with respect to θ, i.e. the set of structures θ such that all the eigenvalues of Δb,θ are simple, is residual (and hence dense) in the set of all compatible positively oriented contact forms on M. In the last part of the paper, we introduce a natural notion of critical pseudo-Hermitian structure of the functional θ↦λk(θ), where λk(θ) is the k-th eigenvalue of the sub-Laplacian Δb,θ, and obtain necessary and sufficient conditions for a pseudo-Hermitian structure to be critical.