Negative numbers for ionic radii?
Ionic radii are not measured on individual ions by some sort of mini-micrometer, but bond distances can be measured and then radii are assigned to the individual atoms/ions. The fuzziness of the assignments can be seen by the variation in ionic radii as the coordination number changes: if you give the ion more room (greater coordination number), it will fill the space - if you assume that the ligands have the same size as usual (whatever that is).
(There are materials (ferroelectrics) in which the coordinated ion does not completely fill the space allotted. In $\ce{BaTiO3}$, the $\ce{Ti^{+4}}$ ion is so small that it can shift its position in the octahedral cage of oxygen anions in response to an external electric field.)
The data in the Shannon paper are extensive, certainly a credit to the Central Research Department of DuPont in 1976, but the trends reported seem to be much more valuable than any specific datum.
Nicolau mentioned covalency as one factor in determining radii. Purepani brought up negative ions not having an effect of negative radii. Assigning a radius to a proton other than zero suggests that you are allowing it to have some electron density surrounding it, able to repel an anionic ligand. Electrostatically, a positive ion will attract some electron body (anion or atom) and come closer; depending on your initial assignment of radii, you can adjust to keep your idea of proton radius constant, or adjust, even into the negative region (because we assume you have a good imagination). Negative ions have a larger size than protons, and are less hard, so they can dent a little. One reason to use negative radii for the cations is that the anion may be dented only at one site, but has a &quot;normal&quot; radius everywhere else around it.
My approach to handling a negative radius is to conclude that the proton (or other mini- cation) is so electrostatically demanding that it can begin to bury itself in the outer shell of the anionic ligand, if the coordination number allows such a deviation in distance to be measured. And since we don't want to redefine the anionic radius, a negative radius is assigned to the proton, and the mathematics works out OK.
I guess you could say atoms and ions are somewhat mushy, and mathematics can deal with it.