In response, Universe Today put a call out to several people with knowledge of those spacesuits that were used in the Apollo 1 mission, which ended fatally in January 1967 when all three crew members died in a pad fire. Howell interviewed Walt Cunningham, Shawn McLeod, field operations manager for David Clark Co., former NASA suit tech Ron Woods, Nicholas de Monchaux, author of "Spacesuit: Fashioning Apollo" and the Smithsonian's spacesuit curator, Cathy Lewis.
A lot of redesigns were made to the equipment to prevent the same situation from happening again, but it appears the flags were not that crucial to the spacesuit design — even though a new spacesuit was used in Apollo 7.
Weeks of searching later, we have some great theories from the experts about why the flags were switched, but no definitive answer. Feel free to let us know if you have heard anything!
As commenters on Universe Today have suggested, the change to the left shoulder (after Apollo 1) may have been to follow the U.S. flag code (about the direction of the flag in travel) but it does not explain why the Apollo 1 crew's suits had the flag on the right shoulder.
The Apollo 1 flag placement may have just been aesthetic, given the use of the American flag as a border on their mission patch, the placement allowed the two flags to be near each other and play off each other — but that is just speculation.
Conjecture aside, a couple of possibly related-points with regards to how the flags came to be on the suits and when and how they were sewn on.
1) The Gemini 4 crew considered the U.S. flag to be their mission patch (they were the first crew to wear the flag). In fact, the crew bought the flags themselves, as Jim McDivitt relates in "All We Did Was Fly to the Moon" by Dick Lattimer.
"Ed White and I used the American flag on our shoulders as our patch. This was the first time the American flag had been worn on a pressure suit and it has continued to be used there ever since. The original flags we had sewn on we purchased ourselves. Later on, of course, NASA made this an integral part of the pressure suit."
Given that it was the GT-4 crew's choice to add the flag, it is not unreasonable to believe they selected the placement too (which in their case was the left shoulder).
2) There was a difference between the U.S. flags worn by the Apollo 1 crew (and earlier crews) and those worn by every crew member since, which could have affected when and how they were sewn on: the material.
The Apollo 1 (and earlier mission) flags were nylon; the post-Apollo fire flags (and all mission patches, name tags, etc.) were silk-screened on non-flammable Beta cloth, which was also used as the outer layer of the redesigned suits.
The Beta patches and flags therefore, had to be specially produced — not something the crew could just purchase themselves — and as such became a more formal part of the preparing the suit for flight.