That's weird if examples works fine.
Try to do debug - put serial.println(dispScreen) in main loop and check if it changes from 0 (main screen) to 1 (menu screen).
Also compare line by line touch screen configuration from example sketches with Stilo, there may be some typo if it's not working with Stilo.
There may be other problem as well - memory leak. I used to have a lot of problems with String objects that were crashing board while compiled on IDE 1.0.2 (but the same sketch was working fine with 1.0.1). The symptoms were similar - the screen stopped responding, but this doesn't explain why it goes to menu screen (all String objects are replaced now with char strings in the sketch).
Maybe some wrong connections or something is causing crashes? Try to put serial.println("1"), serial.println("2") etc in main loop after every function and see which one is causing crash, then do the same in this function to see which command is crashing the board.
Time consuming but there is no proper debug function.