My intuitive understanding of the English language (I am German) would correlate with the following quotation:
The word "press" means to exert force on an object, but no motion of the object is implied.
The word "push" also means to exert force on an object, but there is an implied sense of movement of the object being pushed.
(Source from an online forum)
Therefore, "press a button" is what I consider correct English if the intention expressed is that, e.g., a human hand is lowered onto a button in order to trigger a certain mechanism.
Meanwhile, "push a button" is what I would understand as a button sitting on top of a table when a cat enters the scene and decides to shove it off the table for its own amusement.
An associate corrected me today, stating it would in fact be "push a button" or else the song "Push the Button" would be grammatically incorrect. While I do not take the Sugababes [sic!] as the standard by which I measure correct speach, it made me wonder; is there any rule that makes one of these two options incorrect or are the both of us free to anarchically make use of any of these two choices?