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~~I'm going to run through your list of points and see what happens:~~
~~<ol>~~
~~<li>Fine. Has to interact via either gravity, strong or weak</li>~~
~~<li>Fine. $0$ charge (no electromagnetic interaction), integer spin (are bosons). Think Higgs Boson, photons, gravitons etc. ('force-carriers').</li>~~
<li>"Neo Matter particles have the half life of a neutron". "Up Neotrons have a slightly longer half life (by 15 minutes)" (from comments). I'm afraid you've just contradicted yourself. The half-life of a (free) neutron is just over $10$ minutes. The half-life of a particle comes from the possible decays of that particle</li>
~~<li>Contradicts 3 - if a particle can't decay, then it can't have a half-life (half-life is infinite)</li>~~
<li>"Big as neutrons" contradicts 4 as 'having size' implies 'consists of other particles'. However, this can be adapted to mean that the Compton wavelength is that of a neutron (change as you wish - $\lambda = \frac{h}{mc}$ ), so by definition, is also the mass of a neutron, which is fine</li>
~~<li>Have $0$ charge, so fine</li>~~
~~</ol>~~
~~Ah. Only interacts via gravity. Including that it is a boson satisfies 1 and 2.~~
Now, I'm going to say that your best bet is either to change "cannot be divided into anything smaller" into "cannot be divided into other neo-matter particles" (they could perhaps decay into gravitons, although if they have the same mass, they'd have the same half-life) or get rid of statement 3 altogether.
As the only interaction is gravity, placing 2 of these particles really close to each other (with no acceleration) will automatically cause them to come together to 'form' an atom (more like a binary orbiting system) - molecules form by complex arrangements of this - if you've got, say 1000 clumped 'together' (extremely close), then other clumps of smaller (say 10) would 'orbit' and you have a molecule of a sort.
~~TL;DR~~
Essentially, the gist is that they're fundamental bosons that only interact by gravity, which is an attractive force. If you want them to be different (they have different names, so have to be different in some way), you could give them different masses, which would also give them different decay times. They could possibly decay into gravitons if the conditions are right.
~~Edit to answer questions from comments:~~
The properties are: they have a mass and integer spin (are bosons) and only interact by gravity. This means that the wavefunction of the 'particles' can overlap with no repulsive force, allowing for an atomic 'nucleus' with essentially arbitrary density (depends on the number of up/down neotrons in the nucleus). Outside this, other up/down neotrons (or combinations thereof) will do the quantum version of orbiting this. There's no reason that neotron atoms couldn't be combined with regular atoms as they don't experience repulsive forces, although they'll still have a temperature and so can decay from molecules and atoms back into up/down neotrons. I don't know much about it, but you could potentially get away with using this as dark matter or something similar.
This all also means that the binding energy is given by the gravitational binding energy $\frac{3GM^2}{5R}$ ($M$ is mass of molecule, $m$ is mass of up/down neotron) and intermolecular forces are just that of gravity: $F = \frac{GM_1M_2}{r}$. The interesting bit is that below a critical temperature of $\sim 3.125\frac{\hbar^2}{mk_B}n^{\frac{2}{3}}$ ($n$ is the number of up/down neotrons 'clumped' together), they would (presumably) form a Bose-Einstein Condensate (BEC) that would (presumably) allow us to probe quantum gravity (if only they existed!). As such what happens if a BEC is formed is anyone's guess. Using $M \approx nm$ can be used to give you the approximate amount of energy needed to break apart a neotron molecule (energy required to break apart $\sim$ gravitational binding energy, although it won't be exact as there will be gaps between 'clumps' of particles, which will also be 'orbiting' in some quantum sense of the word, so the actual value will be less than that calculated.)
Interactions: again, the force between a neotron molecule of mass $M_1$ and anything else of mass $M_2$ is $F = \frac{GM_1M_2}{r}$, which is the non electromagnetic interaction. Interactions on the level of quantum field theory (QFT) are completely unknown as it would involve a quantum theory of gravity. However, the world that this exists in could have a quantum theory of gravity (see bit above about BECs), so if you want QFT-level interactions, the best bet would probably to handwave a bit, unless you want to learn string theory or loop quantum gravity or something and use one of those, although decaying into gravitons does seem reasonable to my mind. If you just want classical interactions, then all that matters is $F = \frac{GM_1M_2}{r}$

I'm going to run through your list of points and see what happens:
<ol>
<li>Fine. Has to interact via either gravity, strong or weak</li>
<li>Fine. $0$ charge (no electromagnetic interaction), integer spin (are bosons). Think Higgs Boson, photons, gravitons etc. ('force-carriers').</li>
<li>"Neo Matter particles have the half life of a neutron". "Up Neotrons have a slightly longer half life (by 15 minutes)" (from comments). I'm afraid you've just contradicted yourself. The half-life of a (free) neutron is just over $10$ minutes. The half-life of a particle comes from the possible decays of that particle</li>
<li>Contradicts 3 - if a particle can't decay, then it can't have a half-life (half-life is infinite)</li>
<li>"Big as neutrons" contradicts 4 as 'having size' implies 'consists of other particles'. However, this can be adapted to mean that the Compton wavelength is that of a neutron (change as you wish - $\lambda = \frac{h}{mc}$ ), so by definition, is also the mass of a neutron, which is fine</li>
<li>Have $0$ charge, so fine</li>
</ol>
Ah. Only interacts via gravity. Including that it is a boson satisfies 1 and 2.
Now, I'm going to say that your best bet is either to change "cannot be divided into anything smaller" into "cannot be divided into other neo-matter particles" (they could perhaps decay into gravitons, although if they have the same mass, they'd have the same half-life) or get rid of statement 3 altogether.
As the only interaction is gravity, placing 2 of these particles really close to each other (with no acceleration) will automatically cause them to come together to 'form' an atom (more like a binary orbiting system) - molecules form by complex arrangements of this - if you've got, say 1000 clumped 'together' (extremely close), then other clumps of smaller (say 10) would 'orbit' and you have a molecule of a sort.
TL;DR
Essentially, the gist is that they're fundamental bosons that only interact by gravity, which is an attractive force. If you want them to be different (they have different names, so have to be different in some way), you could give them different masses, which would also give them different decay times. They could possibly decay into gravitons if the conditions are right.
Edit to answer questions from comments:
The properties are: they have a mass and integer spin (are bosons) and only interact by gravity. This means that the wavefunction of the 'particles' can overlap with no repulsive force, allowing for an atomic 'nucleus' with essentially arbitrary density (depends on the number of up/down neotrons in the nucleus). Outside this, other up/down neotrons (or combinations thereof) will do the quantum version of orbiting this. There's no reason that neotron atoms couldn't be combined with regular atoms as they don't experience repulsive forces, although they'll still have a temperature and so can decay from molecules and atoms back into up/down neotrons. I don't know much about it, but you could potentially get away with using this as dark matter or something similar.
This all also means that the binding energy is given by the gravitational binding energy $\frac{3GM^2}{5R}$ ($M$ is mass of molecule, $m$ is mass of up/down neotron) and intermolecular forces are just that of gravity: $F = \frac{GM_1M_2}{r}$. The interesting bit is that below a critical temperature of $\sim 3.125\frac{\hbar^2}{mk_B}n^{\frac{2}{3}}$ ($n$ is the number of up/down neotrons 'clumped' together), they would (presumably) form a Bose-Einstein Condensate (BEC) that would (presumably) allow us to probe quantum gravity (if only they existed!). As such what happens if a BEC is formed is anyone's guess. Using $M \approx nm$ can be used to give you the approximate amount of energy needed to break apart a neotron molecule (energy required to break apart $\sim$ gravitational binding energy, although it won't be exact as there will be gaps between 'clumps' of particles, which will also be 'orbiting' in some quantum sense of the word, so the actual value will be less than that calculated.)
Interactions: again, the force between a neotron molecule of mass $M_1$ and anything else of mass $M_2$ is $F = \frac{GM_1M_2}{r}$, which is the non electromagnetic interaction. Interactions on the level of quantum field theory (QFT) are completely unknown as it would involve a quantum theory of gravity. However, the world that this exists in could have a quantum theory of gravity (see bit above about BECs), so if you want QFT-level interactions, the best bet would probably to handwave a bit, unless you want to learn string theory or loop quantum gravity or something and use one of those, although decaying into gravitons does seem reasonable to my mind. If you just want classical interactions, then all that matters is $F = \frac{GM_1M_2}{r}$

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