Nothing ever ‘pushes’ anything else to any other spot in the sentence in German. Rather, word order is a fine tuning process which determines an overall order that seems most natural or least strained. Thus, please remove sentences such as ‘Haben pushes nicht to the end of the sentence’ out of your mind.
Another issue with your post is what you determined to be the object. The only object in your sentence is das Examen. Am Freitag serves as an adverbial of time much like today or tomorrow but is not an object.
The negation particle nicht behaves a lot like an adverb in German. As such, it can be moved around a sentence rather freely without too much influence of the other fragments. However, it is not able to stand in the Vorfeld — the part preceeding the verb, typically labelled position 1 — alone, which constrasts it against other adverbs.
Sie rennen schnell.
Schnell rennen sie!
Sie rennen nicht.
Nicht rennen sie.
It can be used, however, to negate not the entire sentence but a single fragment of it, as Janka already explained well. If it is used to negate a certain fragment, it is placed immediately before said fragment. If it is used to negate the sentence as a whole, it is usually placed towards the end. And if it is used directly before a noun, it often gets replaced by keine — although note that that is not strictly always true. Compare:
Wir haben am Freitag das Examen nicht. (entire sentence)
Wir werden am Freitag das Examen nicht haben. (entire sentence, but note how the infinitive takes last position)
Wir haben am Freitag nicht das Examen. (Sondern die Exkursion; das Examen ist am Donnerstag)
Wir haben am Freitag kein Examen. (Wir haben überhaupt kein Examen in nächster Zeit.)