As with most questions concerning Rome, it really matters what period you're talking about. During the Republican era, prior to Rome's solidification of control over the entire Italian peninsula, the two groups that qualified were the Sabines and Samnites - stubborn hill tribes that, while difficult to beat in war, were generally seen as a bit 'hick' or 'hillbilly'. The concepts don't translate directly, but they weren't 'civilized' peoples like the Greeks or Romans or Etruscans, but were more demi-civilized.

Later, once control of Italy had become more or less cemented, this viewpoint switched to the Celts, particularly those tribes comprising Cisalpine Gaul. Again, redneck isn't really the best concept here, 'barbarian' is better, but the viewpoint contained many of those same elements that we associate with rednecks today - being ignorant, unlettered, unclean, incestuous, loud, and generally uncultured and proud of it. Read this passage from Strabo to get a sense of how the Romans felt.

After the rise of the Roman Empire, the pattern shifted to one more similar to what we know today, with city dwellers perceiving country dwellers as less sophisticated. This particularly developed during the later Empire, with the rise of Christianity as the official state religion and a sharp divide between those who were with the new hotness, and those poor ignorant souls who hadn't caught on yet. Pagan literally means 'country dweller', and it is a matter of some ongoing debate whether it was originally a term used by Christians to describe other Christians, by Christians to describe adherents to the old state faiths, or if it was a term that predates Christianity in general usage. In either case, the 'rednecks' were those who lived so far in the country that they were literally centuries behind in religious practice.