Why exactly is Neutronium-4 unstable and how to explain Marqués' experimental results? Wikipedia states:

A tetraneutron is a hypothetical stable cluster of four neutrons. The existence of this cluster of particles is not supported by current models of nuclear forces. There is some empirical evidence suggesting that this particle does exist, based on a 2001 experiment by Francisco-Miguel Marqués and co-workers at the Ganil accelerator in Caenusing a novel detection method in observations of the disintegration of berylliumand lithium nuclei. However, subsequent attempts to replicate this observation have failed.

First of all, why is it that the Neutronium-4 particle is unstable? How can we use the current model of particle physics to predict its instability?
And if Neutronium-4 can't exist (in a stable form), how do we interpret Marqués' results? Was it just a mistake? 
 A: 
First of all, why is it that the Neutronium-4 particle is unstable?

The most basic explanation is as follows. The strong nuclear force has a range of about 1 fm ($10^{-15}$ m). Therefore to make a tetraneutron, you need to keep four neutrons all confined to this amount of space. That means their wavelengths have to be about that short, so via $p=h/\lambda$ we find a minimum bound on their momenta and energies. The strong nuclear force does not appear to provide enough negative potential energy to keep neutrons bound when they have this much kinetic energy.
Of course you can get much more elaborate than this, discuss the specific wavefunctions and the details of various models of the nuclear force, and so on, but this is the basic physics.

And if Neutronium-4 can't exist (in a stable form), how do we interpret Marqués' results? Was it just a mistake? 

Yes, it sounds like it was a mistake. Wikipedia gives some fairly detailed discussions of this, with references to papers. The Marqués paper was in 2001, and 17 years is a long time. It seems like the dust has had plenty of time to settle, and Marqués was just wrong.
A: 
First of all, why is it that the Neutronium-4 particle is unstable? How can we use the current model of particle physics to predict its instability?

There is this recent publication that also summarizes recent observations on the four neutron system.The abstract:

We utilize various {\em ab initio} approaches to search for a low-lying resonance in the four-neutron (4n) system using the JISP16 realistic NN interaction. Our most accurate prediction is obtained using a J-matrix extension of the No-Core Shell Model and suggests a 4n resonant state at an energy near Er=0.8 MeV with a width of approximately Γ=1.4 MeV. 

Resonances are observed above the threshold, (in this case of the sum of 4 neutron masses), they have a width and therefore are unstable. At least the link for a nuclear model calculation describes  this. There are calculations using lattice QCD for up to three neutrons, but I have not found for four. Here is one that discussed dineutron possibility of bound states from lattice QCD calculations which take into account the quarks within the neutrons.:  In page 13 B it discusses the dineutron, and calculations depend on the quark masses and nothing is definite:

allow for the possibility of both bound and unbound di-neutrons for light-quark masses larger than those of nature, while indicating an unbound di-neutron for lighter quark masses (45, 46, 47). In contrast, a model-dependent calculation indicates that the di-neutron remains unbound for all light-quark masses (49).

Something  on these lines might be attempted for four neutrons, if experimentally a bound state had been found.

And if Neutronium-4 can't exist (in a stable form), how do we interpret Marqués' results? Was it just a mistake? 

An "honest mistake".  Remember the faster than light neutrinos? Experiments in nuclear and particle physics often are very complicated, and human error can interfere easily. That is why there are two similar experiments at CERN, ATLAS and CMS, to reduce the human error, because the instruments and the analysis are independent in the two experiments.
Back in the 1970's, when we were measuring bubble chamber tracks to reconstruct events and finally get a data summary tape of the four vectors of the tracks for analysis, the film was scanned twice, just because of human error, which is about 5%. This can play havoc if not caught soon enough.
Historical digression on human error: When I started writing programs for computers, back in 1966, we would punch our fortran programs on cards which were in binary code and fed to the computer through a card reading machine. To minimize the human error, the cards were punched twice, the second machine checking that the holes representing the binary code   were correct.
