Hyperbolic Brunnian links and rectangular cusp shapes

The rectangular shapes that you find are probably due to the particular symmetries these links have. When there is a symmetry that fixes a component $C$ of a link which fixes both the meridian and the longitude but inverts the orientation to only one of the two curves, then the shape is rectangular.

This holds because the symmetry induces an isometry on the flat cusp section, which must preserve the unsigned angle between the (geodesic representatives of the) meridian and the longitude, and the only possibility is that they stay at right angles.

Sometimes we find such symmetries when the link is very symmetric and the component $C$ is an unknot: just try to do a $\pi$-rotation that inverts $C$ and see if this gives a symmetry of the whole link. This works for instance for the Borromean rings and the second link you draw.

Added: no, the $\pi$-rotation inverts both the meridian and the longitude... to invert only one of them, mirror the diagram and see if you get the same link up to isotopy, possibly after permuting the components other than $C$. This works for the links $B$ and $N$ drawn above, and maybe on others.

So the reason for rectangular shapes in the pictures above should be "unknotted components and many symmetries".

Any two-component two-bridge link is Brunnian, but most of them do not have rectangular cusp shapes. For a description of the geometry of 2-bridge link complements, you may consult this appendix by David Futer.