Whether you have flats or drops doesn't really matter. With the right size/type of frame, headset spacers and the rights stem length and angle you can get a very similar position with either that doesn't put too much strain on your back or neck. Bar width is also a factor too - riding on the tops of drops gives less width than most flat bars could offer and makes you a bit more hunched, which might not help. I personally prefer wide flat bars with the added bonus that I feel more in control, especially on fast descents, which also reduces tension in my neck and shoulders. There's no right or wrong - just what suits you. It's trial and error - but unfortunately not cheap to put right if you get it wrong!
Riding position isn't the only thing that contributes to a bad back. A stiff frame, hard saddle and thin hard tyres are likely to make things worse. They might be faster in theory, but back pain slows me down more than fat tyres, a comfy saddle and a resilient steel frame do. If you can't keep up with your friends - choose different friends!