Counting the number of thick regions which overlap a square
I think that the maximum number of pairwise disjoint fat regions which overlap the square should be strongly related to circle packing. 

The worst-case shape for a region is something like a "ball &amp; chain". Below I have depicted such a region for $\beta=2$ with diameter 1

.

and these can pack within distance 1 of the unit square obviously much more tightly than I've depicted them. 



Note that the actual ball &amp; chain region is defined by the green area, and the outer circle is just a guide to depict the fact that these regions have fatness 2. In fact, the chain part of the region, can "bend" to allow more regions to be packed.