Due to its position in the sentence (between "Reis" and "Hähnchen", and immediately preceding "mit"), "zusammen" refers to "Reis und Hähnchen". "Mit" always refers to what comes after it (Hähnchen in this case), and "zusammen mit" is kind of a fixed phrase similar to the English phrases "together with" and "along with".
For "zusammen" to refer to "wir", you'd have to reorganize the sentence like this, for example: Wir essen zusammen Reis mit Hähnchen. In that case, because "zusammen" comes directly after "wir essen", it must refer to "wir essen", no matter what comes after it.
It would sound strange to say "Tee zusammen mit Milch" if you were talking about tea with milk in it. However, I think that might actually be a good way of expressing "tea and milk" served together but in separate vessels (as would "Tee und Milch").
For reference, there would be nothing wrong with saying "Wir essen Reis mit Hähnchen."
The order also struck me as odd. In English we typically refer first to the main entree first followed by the side, so it strikes me as more natural to say we are eating chicken with rice, than rice with chicken. I suppose there are some ethnic dishes, such as shrimp fried rice, where you might want to stress that it is a type of rice.
To be honest, "Wir essen Reis zusammen mit Hähnchen" definitely does NOT sound right in German. It makes it appear as though we eat rice in the presence of somebody called Hähnchen. Better would be "Wir essen Reis MIT Hähnchen". Ever better would be "Wir essen Reis mit HÜHNCHEN". Although Hähnchen is the correct term for a YOUNG MALE CHICKEN or the diminuitive form for a male chicken ("Hahn"), you usually eat the female chicken: "das Huhn" or "das Hühnchen".
That would be true if people were more logical. But in reality, the actual gender of the animal being eaten has little to do with the name of a dish, and "Hähnchen" is at least as commonly used in the food industry as "Hühnchen". For example, it would be very rare to find a place selling "Hühnchen-Döner" because they are almost exclusively referred to as "Hähnchen-Döner" in Germany.
Yes, it should be changed to "Wir essen Reis mit Hähnchen."! Because additionally to what amoussa said it could also mean something like "Whenever we eat rice we always eat it with chicken.". It just sounds unnatural. More natural would either be saying just "mit" or "Wir essen Reis und Hühnchen".