How did early estimates of a "potato radius" set 1 eV ~ GMμ/R and get 200 to 300 km?
How do I use this to arrive at a potato radius R of about 200 to 300 km?


You can't, for multiple reasons. One reason is that the authors themselves disclaim this as a valid approach:


  The result is a radius of a few thousand kilometers for a rocky body and ~1000 km for an icy body. This was meant to be only an order of magnitude calculation. We try to improve on it below.


Another is that the authors don't show their math. On trying to replicate it, I get rather different results. The mass of a more or less spherical object is $M=\frac43\pi\rho R^3$, where $\rho$ is the mean density of the object. Plugging this into their expression yields $1\,\text{eV}\approx \frac43\pi G\rho\mu R^2$, or
$$R \approx \sqrt{\frac{1\,\text{eV}}{\frac43\pi G\rho\mu}}$$

This suggests the potato radius is inversely proportional to the square root of density and inversely proportional to the square root of particle mass. This obviously is not the case; the observed potato radius of rocky objects is about 50% more than that of ice objects, even though rocky objects have the greater density and are made of more massive molecules.

I took this as a throwaway paragraph. The key results are further on in the article.