The reason for this is that unlike the electrostatic force the nuclear force depends on how the spins of the two particles are aligned. The force is stronger when the spins are in the same direction than when they are in opposite directions. To see why this causes a problem imagine trying to assemble some number of neutrons into a nucleus. We expect there will be energy levels like the energy levels for electrons in an atom, though they'll be more complicated (this is the nuclear shell model) and each energy level will contain two particles.
We try to put the first two neutrons into the first energy level, but the problem is the strong force wants the spins to be in the same direction and the Pauli exclusion principle doesn't allow this. We would have to flip one of the spins to make the two spins opposite, but this reduces the nuclear force between the particles and it raises the energy of the level. Then we try to add the third and fourth neutron into the second lowest energy level, and we run into the same problem. We can do it, but the energies will be much higher than they would be if the spins were parallel, so the nucleus would be much less strongly bound.
Now this doesn't mean the collection of neutrons wouldn't be bound, but there's a problem. Neutrons freely convert to protons, and you can put a proton and neutron together into a single orbital with their spins in the same direction because they have different isospins. So if you put two neutrons into the lowest orbital with their spins opposite one of the neutrons will turn into a proton. Now the two particles can have their spins parallel, which lowers the energy of the orbital and therefore makes the nucleus more strongly bound. The same will happen with the next lowest orbital, then the next and so on. Your nucleus made up of neutrons would spontaneously convert into a 50/50 mixture on neutrons and protons because it lowers the energy.
This argument implies all nuclei should be a 50/50 mixture of protons and neutrons, and this is approximately right but only approximately. This is because binding in nuclei is more complicated than the rather simple model I've described above. But while the model is wrong in detail it does capture the general principle involved.