When the force applied to the center of the bottom surface, balls shift up and attract neighboring balls. According to this model, even if spring is very rigid, it's unclear, why the whole body is moving. Why do the top balls start to move? Again, body is empty inside; the balls from bottom can't reach the balls from top.
If you have a string, and you push a part of it, it will oscillate back and forth. In your example, instead, the particles affected will move back and forth because they have a changing net force(initially a push forward, then as it moves up the other particles pull it down). As the particles move up and down, they rub off the particles in the air and propel the whole object forward.
If there was no air, then the particles would oscillate up and down forever. It means the affected particles have a changing net force on them, but the other ones do not, and as such they shouldn't move. Mechanical forces cannot act at a distance.
OP you didn't state in the question if the bottomleft particle could collide with the particle above it. If thats the case then yes the object will move forward.