Nails have been used for years and work just fine. The idea of building individual walls with nails then screwing them together will work also as long as each wall is not to heavy to move around later. Screws are stronger than nails so you need fewer of them. But that only helps if you know what nailing schedules are in the first place. Since most people build but just sticking a nail or screw where they think one is needed it is really not that much help. I built my entire coop and run with screws, but I also don't have problems driving them in. My advice on that is practice practice practice. once you get good at it there is no need for pilot holes and you can actually drive a screw faster than a 16 penny nail. Remember the further the screw drives into the wood the harder you have to push. Get the feel for that and your problem with screws will be much less.
I still vote for screws. They are stronger and you can take it all apart again when you realize you made a mistake.
I still vote for screws. They are stronger and you can take it all apart again when you realize you made a mistake.