The sensors could be made to a square format (though the current diameter would not accommodate 36x36mm, it would need to be about 30mm) if there was a demand for it. But by that logic the question we may actually need to be answering is why aren't sensors circular given that lenses present a circular image?
There were some attempts at circular sensors in the phone space when Nokia were working on the PureView cameras like the 808 before PureView just became a marketing thing for their best cameras. The camera then used the surface area it needed to make the chosen aspect.
Going back to squares - other square sensors are available in the 35mm size range outside of the SLR space (like the SBIG stx-16803 aimed at astronomers - thanks DanDan.)
One of the things is that inside a DLSR you also need space for a mirror to move into and also for a shutter mechanism. While the shape of cameras could change to accommodate it, it's a lot of R&D for a small payoff.
It's also telling that even the medium format cameras use a non-square sensor with the Leica S, Pentax 645D, Leaf AFi, Hasselblad H5D and PhaseOne medium-format systems all adopt a rectangular aspect-ratio similar to the 6x4.5 systems than 6x6.
Any increase in sensor area also means a non-linear drop in wafer yield (the number of useful sensors you can get out of a silicon wafer) as demonstrated graphically in this answerthis answer - again it's extra cost.
Little payoff is where the idea really falls down - Canon, Nikon, Sony etc all put huge amounts of money into R&D, if any of them thought there was a viable market then they'd go for it. I'm not seeing one and the lack of such DSLR cameras on the market would back that up.