Continuous-time mathematical formula for deconvolution filters
The FT of the convolutional inverse is the point-wise inverse in the FD. So the convolutional inverse of a gaussian might have numerical problems. The FT of a gaussian is a gaussian. Its point-wise inverse thus can be large in many places.

For a box function, the situation doesn't look all that good either. Its FT is a variant on $\operatorname{sinc}$, which has many zeroes and loci of low magnitude.

You can certainly determine the convolutional inverse of a gaussian, but the inverse of a box function is not even a function. Convolving with a box will cancel any function that is periodic with a period multiple of the edge length.