Objects with temperature 0K are infinitely cold, so it is clear from the definition that they cannot get colder. From a fundamental (i.e., statistical mechanics) point of view, the physically relevant parameter is coldness = inverse temperature $\beta=(k_B T)^{-1}$, where $k_B$ is Boltzmann's constant and $T$ temperature in Kelvin. As $T\to 0$, $\beta\to\infty$, proving the statement.

This doesn't mean that objects with a temperature $<0$K do not exist. However, these are extremely hot and **not extremely cold**! Indeed, if you insert a negative number into the formula for $\beta$ one gets a negative value, which means that the object is less cold and hence hotter than any object with arbitrarily large positive temperature!

On the other hand, ordinary objects cannot have negative temperature; it is a theorem of statistical mechanics that translation invariant systems (i.e., those that can in principle move by arbitrary amounts) must have $T\ge 0$. The only systems that can have negative temperature are those embedded inside a rigid material.

Some references:

D. Montgomery and G. Joyce. Statistical mechanics of “negative temperature” states. Phys. Fluids, 17:1139–1145, 1974.

http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19730013937_1973013937.pdf

E.M. Purcell and R.V. Pound. A nuclear spin system at negative temperature. Phys. Rev., 81:279–280, 1951.

http://prola.aps.org/abstract/PR/v81/i2/p279_1

Section 73 of Landau and E.M. Lifshits. Statistical Physics: Part 1,

Example 9.2.5 in my online book Classical and Quantum Mechanics via Lie algebras.