Water is a good solvent because of it's dipole moment
Water is a good solvent because it has alternating positive and negative charges on either sides of its molecular structure. The highly electronegative oxygen atom pulls electron from the hydrogen towards it, causing a more negative charge on the side of the oxygen atom. This is called a hydrogen bond. Because of the angle between the hydrogen atoms when bonded, this also create a direction where there is a more positive charge due to the two hydrogen atoms. This is called a dipole moment.
Nitrogen is more symmetrical
Diatomic nitrogen (N$_2$), on the other hand, has a strong, covalent triple bond. The bond is 'short' in the sense that the atoms are tightly bound. Due to the nature of a covalent bond, especially one so strong and short as this triple bond, the electrons are shared instead of pulled from one molecule to the other. Thus, there is no dipole moment. Infact, this makes the normally very reactive element nitrogen into an almost inert gas in its N$_2$ form.
Note how, due to the shared electrons in the middle, there are an equal number of electrons, and equivalent electric charge, from both sides of the molecule. Thus, N$_2$ is not going to be a good solvent.
Carbon dioxide is already a solvent
Carbon Dioxide can be used as a solvent in its liquid form, as well. In general, Carbon Dioxide is a specfic kind of solvent called a lipophile. This is basically the class of substances which dissolve in oil, while hydrophiles are the substances that dissolve in water. Just as some things dissolve in alcohol but not water, those same things will generally disssolve in carbon dioxide, too.
Carbon dioxide does not have to be supercritical to be a solvent, but making it supercritical gives it some additional useful properties. On the other hand, Nitrogen is so inert that it is not a solvent as a liquid, so there is little reason to believe it would be a solvent as a supercritical liquid.
There is a more interesting form of nitrogen
Of course, perhaps the most interesting nitrogen compound, from a biochemistry point of view, is ammonia, which is itself a solvent. Ammonium, its ionic form is a lipophilic solvent like carbon dioxide. Its supercritical form can be used as a solvent as well. So perhaps supercritical ammonia or ammonium is a better place to start.