I ... prefer the aspect ratios of 6:5 or 6:6, rather than 3:2 or 4:3. I'd like to be able to shoot directly in this composition rather than crop afterwards.
Square medium format is more or less in the digital world; recent development of medium format cameras is almost entirely in the 4:3 shape of 645 (actually 56x41.5mm), and all recent sensors are also 4:3 shape.
There are still some medium format backs using an older square format 4000x4000 (16MP) 36x36mm Kodak sensor, but that crops significantly from the square MF film frame of 56x56mm. That sensor crop means you are no better of than cropping to square or 6:5 from one of the newer 48x36mm sensors.
I do not understand why someone careful enough to use medium format would insist on a getting one particular shape without need to crop; it seems an amazingly rigid approach to composition. Why not just use an available 4:3 shape sensor, make sure that everything that you want is within the frame, and crop to taste afterwards? Who knows, you might discover that some images work better at shapes other than square or 6:5.
But maybe a gridded focusing screen would help you see where the 6:5 or 1:1 crop will fall?