We present simulation results on the effect of a helium nanodroplet environment on the fragmentation dynamics of embedded molecular systems. The helium atoms are treated explicitly, with zero-point effects taken into account through an effective helium-helium interaction potential. The ionized neon tetramer is used as a model molecular system because, like all the small rare-gas clusters, it fragments extensively upon ionization. All the nonadiabatic effects between electronic states of the ionized neon cluster are taken into account. The results reveal a predominance of Ne2+ and HepNe2+ fragments and the absence of bare Ne+ fragments, in agreement with available experimental data. The neutral monomer fragments exhibit a rather wide kinetic energy distribution that can be fitted to the sum of two Boltzmann distributions, one with a low kinetic energy and the other with a higher kinetic energy. This indicates that cooling by helium atom evaporation is more efficient than was believed so far, as suggested by recent experimental results. Purely classical calculations are shown to strongly overestimate the amount of cage effect (cooling), clearly indicating the need to take into account zero-point effects.