Still can't figue out how anyone can shoot, and hit, with both eyes open. 
Have you ever played any baseball? If you were at shortstop and a ground ball was hit to you, would you close one eye when you threw it to the first baseman?
Shooting and hitting something with both eyes open is not the easiest thing to do without practice (sorry I mentioned it countryboy), but it helps if you have the right equipment. IMO, the right equipment = a peep sight or a low mag scope. If you are trying to shoot with regular open sights you are going to have a hard time shooting with both eyes open. With a peep you look through it and place your front bead on the target (I unscrew the aperature out of my peep sights for hunting conditions); your eye will automatically center the bead in the peep (it is looking for the clearest view of the front sight); this is as similar to shooting a shotgun as you can get.
A low power scope is in many ways better once you get used to it. First of all you need quality glass and a good amount of eye relief, My 7600 has a Leupold Vari-X III 1.5 to 5x; it stays on 1.5 99.9% of the time. I can shoot this well with both eyes open but I also sent the scope back when I first bought it and had the factory install a "shotgun Duplex reticle" which had much heavier (thicker) lines than the standard reticle. I actually received a call from the factory when they received my request asking why I wanted that style reticle in a rifle scope. I explained that my hunting is mainly in northern New England where shots are usually under 100 yards and at moving targets; a heavy reticle helps me to put the crosshairs on a moving deer while keeping both eyes open. The next year Leupold offered this same scope set-up touting it as a "dangerous game" scope; they advertised it as a scope you'd want to put on your African Buffalo gun for when the beast was charging you and you had to have quick sight aquisition... Sounds a lot mor glorious than my idea to pick up running deer easier, but you think they would have sent me at least a free hat for the idea...
Running squirrels are tough enough to hit with a scattergun never mind a rifle... You may not be missing by much at all, but all things relative your target is pretty tiny...
You mention rattling off shots pretty quick; you can get away with that with a shotgun, but with a rifle you still need to aim and this takes time and practice.
I have a .22 pump Remington set up with a low power (1 to 4x) shotgun scope. It's not a Leupold but it is a pretty good quality scope with good eye relief. We went out to the range during the winter when there was snow on the ground. I brought my clay flyer thrower and I set the trajectory so the birds wouldn't fly more than 5' off the ground, much lower than the backstops we have. You'd be surprised how many we were able to hit with the .22 after a while, some on the second or third shot even. With the white snow background you could often see where your misses went as well and we found we weren't missing by much! It was very addicting practice; my nephew thought I was nuts when I suggested it but as we went along even I was shocked how well we started hitting.
So, although you didn't want to hear it, practice is the key, but it has to be the correct practice; one that simulates what you are trying to accomplish in the field. If you shoot off the bench daily, one shot at a time, and then go to the field and shoot 3 shots at a moving target off-hand and don't hit consistantly....well, this shouldn't be a surprise I guess.
