I've read on NASA's page on neutron star that one teaspoonful of that star would weigh over 20 billion ton on Earth. If it was somehow possible to bring it to earth would it: A) Burn and disappear during earth atmosphere entry? B) Assuming we bave 20 billion ton of mass in size of a teaspoon here on earth, would it fall through the ground under its own weight?
The reason that the density is so high is because the pressures are so immense. If we somehow teleported a teaspoonful of neutron star material to earth, it would very rapidly inflate because the pressures aren't high enough to crush it into its dense form. This would effectively be an enormous explosion.
It is difficult to describe what it would inflate out into - the neutron star material can be imagined as an incredibly dense soup of neutrons with some protons and leptons in small numbers. The protons and leptons would make neutron-rich elements like deuterium, but most of the matter would consist of free neutrons. These free neutrons would undergo beta decay to produce neutrinos, protons, and electrons, which would likely recombine to make very large amounts of hydrogen, some helium, and a few heavier atoms. In all of these cases, the atoms would be neutron-rich isotopes, though.
The behavior would look most like a very rapidly expanding gas. It would explode with such force that it wouldn't even need to "fall through the ground" - it would obliterate the floor entirely.