if (buttonState == HIGH)
{
buttonPushCounter = (buttonPushCounter++);
}
Your loop will be running thousands of times per second, so the variable will increment very quickly. You need to detect transitions, not states. That is, you want to increment the variable when the switch goes from HIGH to LOW or LOW to HIGH. To do this, you need to keep track of the switch's last state and compare it to the current one. The StateChangeDetection (I think that is what it is called) tutorial under the Digital section shows you how to do this.
It looks like you have declared a last state variable but don't ever check it or update it.
if (buttonPushCounter = 0); {
All your other if statements look correct, but this one doesn't.
buttonState = digitalRead(buttonPin);{
...
}
These set of curly brackets are unnecessary and distracting.