I'm trying to start with a square, then use Tiled clones to repeat the pattern several times, but I can't seem to find the correct combo of shift and scale settings to get this to work!
I wouldn't use Tiled Clones to create that.
Try this instead:
Switch on snapping options as shown in the example below
Open the Transform panel, and click on the Scale tab. Set the scaling as desired.
Create a square with a black stroke and filled white
Press Alt+D to make a clone
Hit the Apply button in the Scale panel.
Move the square until the corner snaps to the object rotation center of the smaller square below it.
Repeat steps 4 to 6 until finished.