This is a very interesting and challenging question. I'll need to deal with it in several stages.
First, the neutrons you refer to are not themselves stable (lifetimes about 15 minutes) , but let's pretend they are for the sake of the discussion. Let's also assume the neutrons are in opposite spin states so we can ignore the special issues connected with the exclusion principle for fermions. I'm also going to neglect the nuclear interactions between the neutrons, which are very important when their wavefunctions overlap, becaue I think those weren't the issues you were worried about.
Second, the picture you have of things settling into the lowest energy state available is incomplete. It sort of assumes the presence of friction, which isn't applicable on the small scale. Any closed system in a state with some particular exact energy will stay in that state forever. When an atom falls from an excited state to the ground state, it emits electromagnetic radiation. That means that once you include electromagnetic effects, the original excited state wasn't quite a state of definite energy, which accounts for why its appearance can change over time.
For neutral particles, one could in principle lose energy by gravitational
radiation but that process is so extremely slow that it can be neglected for atomic-scale events for the lifetime of the universe. So let's look at what the gravitationally bound states of the two neutrons would be. They'd look somewhat like ordinary atomic states, but enormously more spread out. A quick and dirty calculation indicates a spread of about a million light-years. In order for the neutrons to be at some more or less definite positions (say spread out over a range of 10-6
m or so) the kinetic energy associated with the narrowness of the wavefunction would be by far, far larger than the gravitational interaction energy between the neutrons. In effect, you just have two neutrons spreading out independently following quantum mechanical laws, not noticing each others' existence.
(published on 09/20/2011)