Consider a simple site percolation problem on, for example, a 2D square lattice. Each vertex is randomly either there or not with some probability. If two neighbouring vertices are present, then the edge between them is there too. If any connected component spans the entire lattice, then we have percolation.
Equivalently, one could state the problem as follows. We draw a set of shapes such that all vertices are enclosed, and the boundary of the shapes never cross an edge. We find the smallest possible set of shapes (i.e. such that the largest shape is minimized). If the largest shape spans the lattice, then we have percolation.
Using this, we can then look at different formulations of the percolation problem by placing more conditions on the shapes. Normally we allow the shapes to take any form, and so be quite figure hugging around the vertices. But if we required them all to be squares, I guess the percolation properties would be quite different. In fact, it seems quite clear to me that there would always be percolation in this case. So probably it would be quite trivial.
My question (finally) is: Have percolation problems like this have ever been considered? What are they called and where can I find out about them?
From what i understand, the first idea you mention could be reformulated in site percolation. This has been studied extensively and there are a lot of results for the site percolation of the triangular lattice (dual with honeycomb lattice). The second idea could be close to some continuous percolation model like the 'boolean model' or Gilbert disc model, or eventually the Voronoi tesselation model,both of which are covered in the excellent book of Bollobas & Riordan (last chapter).