int buttonState = 0; // current state of the button
int buttonState2 = 0; // current state of the button
int lastButtonState = 0; // previous state of the button
int lastButtonState2 = 0; // previous state of the button
It looks pretty silly to hang a 2 on the end of one variable when you don't hang a 1 on the end of the other one.
It looks pretty silly to use meaningless names. Why does the switch even exist? The current state and previous state variables should reflect that purpose, NOT the fact that the variable holds a pin state (NOT a "button" state, especially since buttons only have "sewn on" or "fell off" as valid states).
if(buttonPushCounter == 1)
if(buttonPushCounter == 2)
Under what conditions might both of these be true? Mutually exclusive conditions should use else if statements for all but the first condition.
The buttons have no effect
What do your Serial.print() statements tell you is happening?
Why not?
and the servos jitter continuously
Standard question when this happens: How are the servos powered?