I agree with all of the folks who advocate shooting a shotgun with both eyes open. I was 16 when I got a job setting/pulling trap/skeet at a long forgotten place in Elkhorn, NE (Roberts' Shooting Park, part of the Roberts' Dairy: it is now the site of a community college!) in the late 60's, and was taught that concept. No matter which eye is dominant, one "points" a shotgun (as opposed to "aiming" a rifle) and with a bit of practice one will adjust to the sight picture. Back in the day I shot skeet with a Remington 870 12 gauge IC choke with a front bead and a center bead on the vent rib (in "vogue" at the time), and never used either in my sight picture because what I concerned myself with was the muzzle of the shotgun in relation to the moving target (and I admit to have never shot 25 straight in one round). Just prior to pheasant season, a lot of guys would come out to practice their "expertise" and would miss hitting 2 out of 3 clays on the skeet range. As a puller, I would stand directly behind the shooter and could see the shot cloud in relation to the target. These guys were using the bead on the target AND stopping the gun, so most missed shots were above and behind the clay (back in the day, the clays from both high and low houses had to cross a center stake 23 yards from each of the stations (at about 73 mph) except Station #8, which was basically a nearly straight oncoming shot in each direction at the same clay speed).
In my hunting experience, if the rooster did not run down the cornfield row and then fly (a fairly straight-on shot requiring little lead), then one would have to revert to getting the muzzle of the gun way out in front of the bird if a crossing shot.
My longest shot on a bird (pheasant) was when I was home on leave from the USAF in 1972 and had to borrow my Dad's Win Model 12 16 gauge 28" full choke with no vent rib and a VERY small front sight bead, near David City NE and it was stepped off at 65 paces by my Dad's hunting buddy who retrieved it along with his black Lab, Judy. Unless one is using slugs for deer/black bear, or using large shot for turkey, there is no reason to close an eye.
Sorry to be so long winded. Just a 64-year-old fart reminiscing.
Just my $.02 worth.
Jim