If you push the cars' body down, the springs want to push it up again (and down/up/down) just like any (not car)spring would do.
The shocks should stop the up- & down-effect of the springs (= shock-absorber).
A good shock will absorb the up/down movement of the springs that is caused by pushing the car down within one movement. If the car does move up and down several times before the vibration is totally absorbed, the shocks are not good any more...
So the answer to: If there is a vibration in going up and down after pushing down that means your shock absorbers need replacement!! is YES.
with a minor 600+k