The spectrum of quantum dots made from semiconductors such as HgTe and HgS changes from negative gap to positive gap with decreasing size. Furthermore, intrinsic surface states, which are not related to dangling bonds, appear in the negative-gap regime. We investigate theoretically the evolution of the spectrum of HgS quantum dots with decreasing size and show how states evolve from a negative gap to a positive gap as confinement is increased. The lowest confined electron level evolves into an intrinsic surface state with increasing size and, thus, is not derived directly from a bulk HgS band. Due to strong band mixing in narrow-gap semiconductors, spacing between confined levels decreases more slowly with increasing size than for quantum dots made from wide-gap semiconductors. Moreover, dielectric screening becomes nearly metallic as the gap closes. As a consequence, confinement energies dominate exciton binding energies for all dot sizes up to the gap closure. Excitons remain in the strong confinement limit as size increases until the gap closes. Nonetheless, the exciton binding exceeds the single-particle gap for sizes near gap closure, opening up the possibility of an excitonic insulator phase in quantum dots not possible in positive-gap quantum dots. Signatures in the quantum-dot optical response for gap collapse and surface states are identified.