This can be done natively.
If the updates are being done on a form and not in Grid Edit, what I suggest is to create fields called [Push Count] and [Next Push Count] The first is just a numeric field and the second is a formula field of [Push Count +1]
Then a form rule that says
When the record is saved and the [Task Due Date] has changed
Change [Push Count] to the value in the field [Next Push Count]
you may need to have the [Next Push Count] field on the form to get that working. Once its working you can hide that field with a form rule.