Probably. I was using the figures in the OP, being too lazy to confirm them. But the precise gravitational field at the surface of a uniform-density cube is a nontrivial calculation, so I’m not going to get an exact number (I was using a spherical approximation, which is why I left so little precision in my answer).
Yeah, I used a sphere of radius .62 mm to give a volume of approximately 1 mm[sup]3[/sup], too. I figure if you had a cubic mm of neutronium, the pressure of gravity would force it into a sphere anyway.
Well, you could build a tiny little model of the planet-eating ship from ST:TOS “The Doomsday Machine.”
If the surface gravity were 1700G and you found a way to contain it, it’d still be plenty dangerous through tidal effects alone. I’d imagine if the pressure weren’t enough to cause it to fall through the earth as if it were nothing, the tidal effects would cause it to sink gradually(er) through deformation like a heated cannonball through a block of ice (I’ve never seen that, actually, to be honest, but you get the picture.)
A guy we had on the SDMB for a few months, Bill DeSmedt, wrote a book called Singularity where a rogue group left over from the breakup of the KGB is doing something suspicious with a teeny tiny black hole that’s orbiting inside the earth (from the Tunguska event). One cubic millimeter of neutronium and a tiny black hole are a bit different, of course, and of course it’s fiction, but he had a lot of help from a couple physicists in writing the technical details, and it might be of interest to people who think about stuff like this.
What would happen to the stuff once it is outside the star? What would it form as it becomes more conventional matter - atoms, energy, particals, ice cream - what does this stuff become?
You could use it to knock starships out of hyperspace & start a career as a space pirate. (Dangit, which Niven story was that? I think that was a micro black hole, but same principle, really.)
My best guess would be either hydrogen or iron, but I haven’t been able to find anything definitive (and not for lack of trying: I’ve asked all of the neutron star specialists here, and they don’t know either).
“The Borderland of Sol.” Niven wrote that right after Jackson and Ryan came out with their primordial black hole hypothesis for the Tunguska event, which fits neatly with the book I mentioned above.
In another Known Space story, Protector, there’s a sphere of neutronium held together with a magic gravity machine. See how it all ties together? 
That means it’s brittle, not weak. When dealing with something with multi-thousand gee surface forces, you’ll want something hard, otherwise it’ll just rip loose chunks and burrow right through. Diamond, apparently, isn’t enough.
After thinking about this, and the Borderland of Sol, it seems to me what you want is to spray it with iron, which will be held on by the thing’s gravity, or give it an electrical charge ( let’s use an ion engine, just as a homage to Niven 
) and use a really, really, really strong magnetic field to lift and move it. Assuming that whatever magic that’s holding it together doesn’t fail, of course, or crush the iron into more neutronium, or bounce the electrons.
This is why I love google calculator (which happens to agree with you on the 88,000 number). 
If it were iron what a shame, I guess since it came from iron as the final stage as a star, it would indicate some ‘atomic memory’ effect, or a different state of matter of iron that we stil don’t know about.
In some ways it seems like anything could be created if the neutrons ‘break’ back apart, perhaps it would just be free neutrons however.
But if they did break apart any reason this stuff can’t form antimatter instead?
I don’t see how; it’s neutronium, not antineutronium.
190068000 lbs. Or, 13857630 stone depending on your preference in antique mesures.
And a cubic mm of nutronium would be spherical, assuming it was not a rapidly expanding sphere, for some magical reason. (Well, given sufficient magic, it can be shaped like Cthulu, or Brittney Spears.)
Tris
Iron because it’s the lowest energy per nucleon, so if the explosion is slow enough, it’d have a chance to settle down into that state. It’s no more memory than is the fact that the puddles form in the same place after each rainstorm: The puddle doesn’t remember being wet, it’s just the low spot. If it’s faster than that, though, it’d be scattered into individual free neutrons, which would subsequently decay into protons and electrons (and neutrinos, but who cares about them?). The protons and electrons would then settle down into a very hot cloud of hydrogen gas.
It couldn’t decay into antimatter, since baryon number (a property of protons and neutrons) is conserved, and since neutronium has a whole bunch of neutrons (and protons, for that matter), it’d have a very large positive baryon number. To get any significant amount of antimatter out, you’d need to convert that positive baryon number into a negative one, which isn’t going to happen.
You might be encouraged to learn, though, that the situation is different for black holes. Black holes don’t care at all about baryon number (nor about most other “conserved” quantities in physics), so when a black hole evaporates, it’ll show no bias towards matter or antimatter, and produce approximately equal amounts of each (no matter what the hole was made from in the first place, presumably matter).