The first mention of a flag for the Land of Oz appears in Dorothy and the Wizard in Oz, in which it's described as such: "This flag was divided into four quarters, one being colored sky-blue, another pink, a third lavender and a fourth white. In the center was a large emerald-green star, and all over the four quarters were sewn spangles that glittered beautifully in the sunshine. The colors represented the four countries of Oz, and the green star the Emerald City."
I can see sky-blue for the Munchkins and lavender for the Gillikins, but why would the red Quadling Country be represented with pink and the yellow Winkie Country with white? Indeed, when L. Frank Baum next describes the national flag in Magic, he writes, "From a hundred towers and domes floated the banners of Oz, which included the Ozmies, the Munchkins, the Gillikins, the Winkies and the Quadlings. The banner of the Munchkins is blue, that of the Winkies yellow; the Gillikin banner is purple, and the Quadling's banner is red. The colors of the Emerald City are of course green. Ozma's own banner has a green center, and is divided into four quarters. These quarters are colored blue, purple, yellow and red, indicating that she rules over all the countries of the Land of Oz." This description makes more sense, and it's the model that's been used for most Oz flags I've seen, including the ones that came with copies of Glinda. They also incorporate the O-with-a-Z-inside symbol that John R. Neill presumably created. Here's one such flag, although the picture is upside down:
For what it's worth, I've heard that Baum's original manuscript used "Ozmites" instead of "Ozmies" for inhabitants of the Emerald City, but neither term is used elsewhere in the series. Getting back to the flag, one interesting suggestion I've seen was that the banner that shows up in Dorothy and the Wizard was an old one with faded colors, which would explain why they're all so light. In Captain Salt, there are Oz flags that are simply solid green with the Oz symbol in the middle, which sounds like a better description for the flag of the Emerald City (or the banner of the Ozmies/Ozmites, if you prefer) than for the nation as a whole.
The books never pay much attention to the flags of the smaller countries within Oz, and the implication in Magic seems to be that they're simply solid in color. In the 1987 Oziana, however, editor Robin Olderman held a contest to design a flag for one of the colored quadrants. The cover of that issue featured Olderman's own design for the Quadling flag, as drawn by Melody Grandy.
The winner of the contest with this one for the Munchkin Country.
Since I haven't seen any other attempts at quadrant flags (I wonder if any of the old contest entries still exist), I don't see any particular reason not to accept these two. What the Gillikin and Winkie flags look like, I couldn't say.