Sheepdawg,
I agree with Frog4aday = "Eye dominance is not an issue ... ." Plus, Cody's view to "absolutely let him shoot left handed ... " ignores the reality of one size does not fit all. Is your 11 year old a true, 100% southpaw? Agreeing with Frog, I believe it's possible he can be a lifetime RH shooter. Does he have a RH or LH preference with rifles?
I don't think you quite interpereted me correctly, I did not have a one size fits all view by any means, my son is right handed but must shoot left handed in order to see, because of eye issues.
Myself, I have actually never been able to consistently pick a dominant eye, even when I went to the eye doctor to get lasik eye surgery, the doctor was almost frustrated with me, because when we did a standard dominance test, I kept coming up with different eyes, and did not have a consistent winner. However, if I do 10 tests, I am usually evenly torn, but if anything MIGHT favor my right eye, but am left handed with handguns and rifles, I throw a baseball right handed, write with a pencil left handed, and shoot a bow right handed. That last one is funny, because I was 4 years old and wanted to learn to shoot a bow, but my father was away on work, and my mother taught me, and couldn't figure out how to teach me left handed, so she taught me right handed, so therfore I am right handed now ha ha. That certainly isn't one size fits all though, ha ha!!
In a young shooter, they are generally very moldable, if they are not too dominantly right or left handed, I would generally let their eyes pick. If they don't really have a dominant eye, let their handidness pick. If they are like me, and aren't solidly left or right handed, as well as not left or right eye dominant, just let them pick, I suppose. However, if someone is extremely right handed, but can't get their right eye to work, then must be left handed, as eyesight is needed to shoot. If they simply can't seem to function with the side they are dominant with their eyes after a distinct effort, but can still shoot well with cross dominance, let them be. My father is cross dominant, and a very good shot.
My approach is anything but one size fits all, it is likely more fluid than most, I outlined that in less detail in my initial post. He said his son is left handed, and left eye dominant. Why would you even consider switching him? Let him be left handed, and just have the kid learn to use a right handed rifle, if procuring a left handed youth gun is challenging. Then when he gets to an age where he can shoot a full size rifle, switch him to a left handed gun.
I started with a right handed guns, as guns are expensive, and we didn't have the money to buy ones specifically for me to learn. When I was about 8, I got a left handed savage youth .22, then a youth .410 sxs, then a left handed ruger 30-06 full size rifle when I was 11, and used that to kill my first deer. However, I almost purchased a right handed rifle in favor of a .308, we couldn't find any left handed .308's at the time other than a remington sendero, but my dad urged me to go left handed. My next rifle purchase was a right handed 22-250 varmint rifle, then a right handed .17 HMR, then a right handed .308 win (which I still have, but instead chambered in .260AI), then a right handed semi auto shotgun....as a matter of fact the first left handed gun I purchases after I was 11 was 18 years later, when I built myself a custom, and I have never had a hitch.
This is why I urged to have his son learn right handed rifles as well, if you practice with both, in this generally right handed world, being left handed, you will be more adaptive to whatever platform you are using vs. being hindered by only knowing one. From personal observation, when many right handed shooters try to shoot a left handed bolt gun fast, they look clumsy and awkward, and can'tdecide the best way to do it. Hand a well adapted left handed kid a right handed bolt gun, and they will be nearly as fast or faster than a right handed shooter, especially prone from a bipod or other front rest.