Let's start with the physics.
1. Gravity acts to exert a force that attacts two objects together. Practically speaking, the force on us is always downwards, toward the center of the earth.
2. To keep an object upright, an equal force must be keeping it from falling down toward the earth.
Imagine two people, one sitting upright and the other standing upright. In both people, at the level of the pelvis (hip), the weight of the entire abdomen, torso, upper limbs and head must be supported by the pelvis. The amount of force acting on the pelvis is the same no matter if the person is standing or sitting because the weight above that point has not changed. Assume, for the purposes of this example, that there are no other supports for any other part of the body other than the seat itself (no armrests for example).
This idea of force being transferred to the ground through the legs and therefore not being put on the pelvis is pseudoscience.
I'll leave the issue of squatting to you, the reader, to figure out. The only thing I will say is that I think those kids in the playground were squatting because there was nothing nearby for them to sit on.
There is some truth to the claim that slouching and bending will add extra stress on the vertebral column, but again, this extra stress will arise whether the person is standing or sitting while slouching/bending. It is the slouching or bending that causes the problems, not the sitting down.
Sitting may encourage some people to assume a poor posture but it does not always follow that all people who sit down have a poor posture.
By the way, anyone who wants to eschew sitting because of believing in this pseudoscience is quite welcome to do so - it means there'll be less seats taken up, making it more likely I'll get one when I next take the subway.