The effects of electron-electron interaction on the electronic structure of a three-dimensional model quantum box are studied. The potential in the lateral plane of the quantum box is taken to be a two-dimensional harmonic potential, and an infinite quantum-well potential is used to represent the confinement in the vertical direction. Three-dimensional two-electron wave functions are constructed by the configuration interaction technique. Exchange and correlation are found to affect the electronic structure strongly. The effect of the finite thickness of the quantum box on the electronic structure is examined. It is shown that the electron-electron interaction can induce interesting polarization effects not only in the lateral plane but also in the vertical direction of the quantum box. Numerical results based on GaAs material parameters are presented.