If the quote refers to consciousness as "awareness", then as pointed above, I would strongly disagree this is a social construct. However I would prefer to look at the quote from a "nature versus nurture" perspective, in which Skinner is arguing for the extreme nurture side, by stating that our consciousness, in this case defined in terms of behaviour, language, and perhaps knowledge (shared knowledge) is a product of society and therefore of the way we were raised. Interpreted in this way it might not appear as outrageous of a statement as the first interpretation.
Regarding the computer network example above, it is an interesting point to make, and it becomes very difficult to pin exactly what is meant by consciousness, I think we only really know that we are conscious, as pointed out by Descartes' "Cogito ergo sum" I think therefore I am, and by induction we assume that other people, given they act, speak and react similarly to us, must have a consciousness as well, however this assumption cannot be proven logically, it is taken upon faith and "common sense". It becomes even more interesting when we consider the computer system "consciousness" possibility, and more importantly the "Zombie" possibility, where people around us act, speak, and do the same actions as us, but do not in fact have a consciousness, it is impossible to distinguish from a "zombie" and a normal human being (this is a mental exercise, I am not realistically suggesting zombies). However zombies require the acceptance of something known as "Cartesian dualism", the belief that mind and body are separate things, it would be impossible to have a zombie with an identical body but a different mind, if the mind was present physically in the body, the difference would be observable.