I’ll start by clarifying that I don’t actually have a need to make square holes at the moment but I am facinated by these techniques.
The maker of the above video Jacques Maurel also wrote an article for Model Engineer a couple of years back. The article outlines a home made square drill using a tool shaped as a Reuleaux rotor.
Jacques article includes how to machine the rotor, jigs for making the parts, grinding the tool and how to make the floating chuck. The floating chuck is based on an oldham coupling and can also be used for reamers and tapping. As seen the first video the solution works well and can produce blind square holes.
Model Engineer magazine issue N°4285 oct /nov 2006
Also see article by Mr Parkes in Model Engineer N°112
Applications of shapes of constant width by John Bryant and Chris Sangwin
An alternative solution to producing a square hole is to use a broach which are good for though holes, keyways and even internal gears.
Michael Cox has written an interesting article for Model Engineer #185 (Jan 2012) where he has a free running rotary broach that works by being just a couple of degrees off of perpendicular to the lathe axis. It uses a single ball bearing as a thrust bearing based on Edgar Westbury’s rotating centre design.