One of the reasons why not is that neutrons are unstable, decaying into protons. They only survive in nuclei because their energy is effectively reduced in that environment. But there is another island of stability, which does indeed comprise only neutrons. In a neutron star the force of gravity generated by a huge number of particles crushes everything together into an enormously dense mass. The electric charges are sent packing in the form of electrons - in effect the protons have decayed into neutrons.
Surface tension plays no part here - the surface is a negligible part of the whole.
Neutron stars are like no other object. Nature seldom repeats herself exactly. If the symmetry is broken, drops are no longer spherical. They also have mirror symmetry because the threads have no preferred direction. For water drops resting on a horizontal surface, gravity breaks the up-down symmetry, although the drops are circular in plan.