How do you become a 240 or SAW gunner in the US Army?

Those systems aren't exclusive to infantry units, and many line support units have them, as well. You can ask to be assigned to those weapons systems, but, especially in the case of the 240, you'll have to get some experience and demonstrate some proficiency with it. In a line unit, for a 240, you'll typically have to start off as an ammo bearer or assistant gunner, then work your way to the 240. For the M249 (SAW), it tends to be more luck of the draw. Do a 25 miler, and you won't be so crazy about either system, I can guarantee you that.