Well, mathematically, k-means clusters are not spherical, but Voronoi cells.
However, the claim is not invalid, as the actual data usually does not fill the whole cell, but if you'd take the convex hull of the data it indeed is somewhat spherical in nature.
The reason probably is that when minimizing variance (and k-means minimizes the in-cluster variance, aka: sum of squares) you do also minimize euclidean distances: the squared euclidean distance is the sum of squares. And since the square root does not change ordering (it's monotone!) the assignment rule of k-means definitely prefers spherical clusters, by implicitly preferring Euclidean distance assignment.
Yes, k-means can be changed. Use k-medoids/PAM with maximum norm then (don't just exchange the norm - you may lose convergence. K-medoids/PAM is guaranteed to converge with arbitrary distances!)
Still, the result will not enforce a rectangular shape of clusters. They may still overlap in unexpected ways. The result will likely look like this (actually, rotated by 45 degree, but obviously this does not change the nature much - a strong preference for 45 degree angles):