This looks a bit ugly, but maybe if we think a bit more about what a zero resistance wire is, we can work out why we won't get something physically unrealistic.
One way to get a zero resistance would be to use superconductors. These are very weird materials - they have huge quantum effects, but the Johnson-Nyquist noise theory you're using in your question is semi-classical, so we might reasonably expect it to not work when lots of quantum things are going on.
In fact, in a superconductor, there are two conducting 'fluids' sharing the same space. One, the normal fluid, is made of electrons, and acts like a electrons in a normal material. This will have thermal fluctuations just like the ones which cause Johnson Nyquist noise. The other, called the superfluid, is made of cooper pairs, and has zero resistance. So it will short out any external current or voltage (which is what makes superconductors into perfect conductors). But it will also short out the noise voltage from the normal fluid. Every thermal fluctuation in the material will be immediately and completely cancelled by a movement in the superfluid, so there will be no Johnson-Nyquist noise. There may well be other noise, but that's a whole other topic.
That leaves us making a zero resistance wire from normal materials, which is of course impossible. So the problem isn't that the current is infinite, it's that it tends to infinity as we reduce the resistance. To see if that makes sense, we have to think about what reducing the resistance to zero really means.
The resistance of a block of material is a material-dependent constant times the length divided by the cross sectional area. The two ways of getting zero resistance are then:
To increase the area to infinity. Having infinite noise current in an infinite area seems reasonable, the current density is the same as it would be for a finite block of material.
To reduce the length to zero. This one is a bit trickier, and I'm not sure my solution is correct. But I think this boils down to a geometry thing. If the loop circumference tends to zero, then the thickness of the wire must also tend to zero, or it isn't a loop of wire anymore. This means there is a minimum resistance, where you can reasonably apply Johnson-Nyquist theorem. Beyond that you have a plate of copper with a hole in, and you'd have to analyse that differently. There is a whole sub-field of physics called fluctuational electrodynamics and you'd probably find the detailed answer somewhere in there.