For those who know any French, the word for "egg" in French is "oeuf," which is pronounced roughly like "off." In Hebrew, "chicken" is "עוף," roughly pronounced like "off." Because Biblical Hebrew came before French, this confirms that the Chicken "עוף" came before the egg "oeuf." Like this comment if you get my joke. (The chicken came before the egg, proven by languages.)
You can argue it's correct. But in the meaning of "bird", "עוף" is outdated. It's used in some set phrases such as "עופות מים" (water birds) and "עוף דורס" (bird of prey), but generally "bird" is almost always "ציפור", and "עוף" is almost exclusively used for chicken meat.
I think it's safe to say that most of the time you can change the order. However, sometimes we use the word order to stress different parts of the sentence. For example, יש לנו עוף(the basic form of the sentence) stresses the chicken, or the 'have', depending on the way it's said and the context. לנו יש עוף stresses 'us'.