We study the contribution of confined and interface phonons to the polaron energy in quantum-well wires. We use a dispersionless, macroscopic continuum model to describe the phonon confinement in quantum wires of circular cross section. Surface phonon modes of a free-standing wire and interface phonon modes of a wire embedded in a dielectric material are also considered. Polaron energy is calculated by variationally incorporating the dynamic screening effects. We find that the confined and interface phonon contribution to the polaron energy is comparable to that of bulk phonons in the density range N=105-107 cm-1. Screening effects within the random-phase approximation significantly reduce the electron-confined phonon interaction, whereas the exchange-correlation contribution tends to oppose this trend at lower densities.