I know that currently free quarks do not exist but can they exist under any circumstance? The Physicsworld article Quarks break free at two trillion degrees states that they do but I want a knowledgeable, second opinion.
Secondly, if they cannot exist freely how can they be considered a particle, far less a fundamental particle?
If you mean can quarks break free of their confinement by the strong force, then the Quark-gluon plasma article on Wikipedia says (cutting and pasting bits here):
Quark–gluon plasma is a state of matter in which the elementary particles that make up the hadrons of baryonic matter are freed of their strong attraction for one another under extremely high energy densities. These particles are the quarks and gluons that compose baryonic matter. In normal matter quarks are confined; in the QGP quarks are deconfined.
Quark–gluon plasma filled the entire Universe before matter as we know it was created. Theories predicting the existence of quark–gluon plasma were developed in the late 1970s and early 1980s. Discussions around heavy ion experimentation followed suit and the first experiment proposals were put forward at CERN and BNL in the following years. Quark–gluon plasma was detected for the first time in the laboratory at CERN in the year 2000.
For sources, see the article itself. There is also more recent work at CERN.
A quark cannot exit a hadron except when it has sufficient energy to create a quark-antiquark pair, leaving the newly created quark behind in its place. There's no sense in which a trillion-degree universe changes this basic fact. It's just that in such a universe, every particle has sufficient energy to create quark-antiquark pairs all the time, so you stop noticing the restriction.
The question of whether a confined particle can be considered fundamental is a very tricky one. We prefer to measure particles that are freely moving, because that is convenient to our detection capabilities. Our theories were developed assuming free particles, because that makes the math easier. Unfortunately for experiment and theory, the universe appears to require fundamental particles that are confined. Since fundamental-ness means "has no more-fundamental internal particles", being free is not required to be fundamental.
Quantum chromodynamic (QCD) as the theory of strong interaction has some important features. Two of them are confinement of color and dynamical breaking of chiral symmetry. At sufficient low energy (or equivalently low temperature) chiral symmetry is broken and colors are confined. The general picture of QCD phase diagram are still on debate specially at finite baryon density. However, at zero chemical potential there is a good evidence from lattice gauge theory that shows a cross-over transition from hadronic degrees of freedom to plasma of quarks and gluons. People at RHIC and LHC have shown such matter produced after collision of nuclei. Maybe the first few pages of this paper helps you to gain some ideas.
