Is there a possibility that an older child could push the button and make the Jackey fold when little sis/bro sits on the Jackey?
Actually, no, it can't. The stroller is made so that it folds forward - that means if a child is sitting in the pushchair, it's practically impossible to fold it. For the auto fold to work, you need to kind of "throw" the handlebar forward which, again, with a child sitting inside is not possible.
Also, the folding button consists of two buttons actually, one is the main middle button and one is hidden underneath - it is in the form of a sliding mechanism. It is rather difficult to smaller children to activate both at the same time.
So, to sum it up, it should not be possible for a child to fold the stroller down by itself with another child inside. That said, it is of course always advisable never to leave the child unattended in the stroller.