I was able to use the MATLAB command line to change the padding around the image.
figure = openfig("example.fig")
This will save all of the properties of example.fig to figure. You can then look at some position values to get location and size of the image area specified as a vector of the form [left bottom width height].
A couple of theses are Position, OuterPosition, InnerPostion. I am using Position for this example. I want to first get the current info:
I then want to set the width and height:
figure.Position = [600 600 200 200]
This preserves the plot dimensions and sets the padding to 200 pixes on each side. You can see the before and after:
This will still be not quite suare since the width and height values do not take into account the title space, but the borders are reduced.