Is this possible without motion tracking? I want to place something at a specific point that is not a pin. The mesh also bends.
Is sounds like you've already found the "Puppet Pin" Position parameter and that that's not getting you the coordinates that you're looking for.
Have you tried hovering the mouse over the square that you're looking to get the position information for while viewing the Info Panel?
If there's an exact spot you're looking for on the square, you could create a duplicate footage item and paint the spot on the square with a color that stands out. Track that dot and then replace the artwork with the original art (the source footage that does not have the spot).
If i have a shape layer with several small squares in it, which has a puppet tool on the layer, how do it determine the position of one of the squares after the puppet tool has moved them?
Aside from this sounding like the most awkward and terrible workflow one could imagine, there is no real way to determine the actual position. Using the pin positions might give you some rough idea, but it's not the exact position of the actual pixels, only the input data for the deformation matrix.