Can stars contain neutron-star dense cores? purely for curiosity:
are there any conditions under which a small amount of neutron star matter (400 trillion grams per cubic centimeter) could be at the core of a star or planet?
neutron star matter would have to exist naturally in small chunks.  perhaps a collision of a white dwarf and a neutron star could mix a large piece of the former with a small piece of the latter; and ordinary matter could form the outer core while the neutron matter could form the inner core?!
or would the neutron matter deflate back into ordinary matter no matter how much ordinary matter surrounds it?
if no, then same question about white-dwarf matter.  could a little piece of it form the core of a star or planet?
 A: In order to produce neutron-degenerate material, you need to compress matter to roughly $10^{15}$ kg/m$^3$. Unless you do this, then any free neutrons will decay. Thus there does not exist a stable, low-density form of partially degenerate neutron matter. 
Stars of course do contain such reservoirs of mass, that in principle it seems that neutron-degenerate material could exist at their centres, with their gravitational field providing the required compression. The problem is however that the cores of stars are not cold.The pressure in the core, required to support the weight of the star, is sufficient to keep the gas at densities far too low to be neutron degenerate. This pressure and temperature can easily be sustained in non-degenerate conditions by nuclear reactions. It is only when exothermic nuclear reactions cease and (neutrino) cooling mechanisms become efficient, that the core can collapse to the required densities at the onset of a core-collapse supernova.
Electron degeneracy as found in white dwarfs requires much lower densities. These conditions are expected to be present in the inert cores of low mass red giant stars, and the cores of brown dwarfs and giant planets that do not sustain nuclear fusion.
In both cases, "small chunks" cannot exist in isolation, because their internal kinetic energy density would be vastly higher than their surroundings. That's a fancy way of saying  it would explode. The minimum mass for a "chunk" of stable neutron-degenerate matter is about 0.15$M_{\odot}$. It is smaller for electron degeneracy (e.g. core of a giant planet), but I do not have an exact number.
Further reading: What is the theoretical lower mass limit for a gravitationally stable neutron star?
What would happen to a teaspoon of neutron star material if released on Earth?
A: One cannot go and look at a neutron star, but one can model with the known physics what the surface of a neutron star would be. Because you are talking of a star it does not mean that it will be luminous. It seems that the models give a very hard crust:

To find out how strong the crusts of neutron stars really are, Horowitz and a colleague created a computer simulation of a star's surface. Though the interior of the star is a kind of fluid mass of mostly neutrons, the crust is composed of broken-up atoms, the nuclei of unknown elements. To simulate this, Horowitz used the computer program to squeeze together virtual selenium atoms, pressing them into tiny cubes. He determined that the crust is billions of times stronger than even the hardiest metal alloys here on Earth.

It seems that strong gamma ray bursts have been connected with neutron star quakes:

In 2004, astronomers spotted a spectacular gamma-ray explosion bursting off a neutron star in the Sagittarius constellation, 50,000 light years from Earth. The star, SGR 1806-20, is a magnetar, a type of neutron star that has a powerful magnetic field.? NASA and European satellites and astronomers around the world detected the flare, which for a tenth of a second was brighter than anything ever seen beyond our solar system. It was the biggest such flare ever spotted and one of only four that have been seen so far.
"We think that these giant flares are coming from really, really big star quakes," said Indiana University physicist Charles Horowitz. Only a super-strong crust could have exploded so forcefully, he explained.

Would a broken off bit of the crust remain in this state if the neutron star broke up? Evidently not, because it is pressure that keeps it as a solid. In space by itself it would expand and revert to the appropriate state of matter for that temperature and gravitational pressure.
A: I'm sure you know why neutron star is so dense and its molecules are tightly packed ?
Because of the presence of very high gravitational field surrounding the neutron star, so the void space that is normally present between atoms gets compressed and no space is left between them.
The matter from the neutron star will surely deflate back until or unless there is higher gravitational field surrounding the neutron star matter.
That can provide it with the same conditions as before.
A: In his book "The physics of superheroes", Dr. James Kakalios argues that the surface gravity of planet Krypton, homeworld of Superman, is about 15 times the value of the surface gravity on Earth.  This is only possible if the deep inner core of planet Krypton contains a neutron star.  The relevant discussion can be found online through GoogleBooks, at page 42.
Obviously Krypton is not a star, but there you have it: there are conditions under which this is possible, at least for a planet.

[c'mon guys... cheer up!]
