It can't be solo neutrons, because they are unstable and decay into protons.
So far as we know, there's not a stable configuration of mostly-neutrons that occurs in nature intermediate between heavy nuclei (uranium is roughly 3-to-2 parts neutrons) and neutron stars of 1-3 solar masses (which are about 90% neutrons).
What you're describing would be the kind of dark matter called a "MACHO," or "massive, compact halo object."
Thanks to recent gravitational lensing studies, where a robotic telescope continuously watches many stars to search for brightening due to the gravity of an intervening dark object focusing extra starlight on Earth, we now have a census of these items down to about the mass of Jupiter. The planet-sized MACHOs outnumber stars by about two to one, but only contribute a few parts per hundred of the total mass of our galaxy. The "dark" contribution of the mass of our galaxy is a few times larger than the luminous mass.
There's actually a pretty firm estimate of the total density of protons and neutrons (collectively, "baryons") in the universe, based on the chemistry of what's out there. Most nuclei are ordinary hydrogen; about 25% are helium-4; various tiny fractions are deuterium (heavy hydrogen), helium-3, and lithium-6 and -7. We know an awful lot about how those light nuclei interact with each other from accelerator experiments, and so we have a very convincing model of how much of each species should have been produced during the Big Bang. Furthermore we can say how many photons should have been produced per nucleus: if there were much more or less than $0.6\times10^{-9}$ baryons per photon at the time of the Big Bang, then the light-element chemistry of the interstellar medium would be measurably different than what it is.
Most sensible people are reluctant to say "the invisible stuff that makes up the bulk of the gravitating mass of the universe must be a fundamental particle that we've not encountered yet on earth." But the case for that scenario is actually quite strong.
