That cannot be all your code? You should use "INPUT_PULLUP" as pin mode for the buttons and test them for being "LOW" and you should "debounce" the buttons.
You code is waiting for the first button (button_on) to be pressed. After that it enters a while loop where "button_on1" and "button_on3" is tested for changes. But those two last values does not change in the while loop and that is why it does not work 