No wonder you were confused. The phrasing "if women were a country" is a little awkward. Or, to be more charitable, it's using poetic license. This is not a way of phrasing a sentence that you are going to come across often, and you might even only see it in this one case.
What the phrase is essentially saying is, "if women had their own country." From that, more indirectly, we can also take it to mean, "if women were in power in a country." Then we can go even one step further, to interpret it as, "if women had equal power as men in a country."
Thus, if women had enough power or equality in Japan, then the women's soccer team would be recognized for their greater degree of success than the male team, and would be treated at least the same. They would fly first class like the male team does.
The point is a little less clear when the author talks about Candace Parker, who I'm not familiar with. I'm only going by what's said in what you quoted.
From what is said in your quote, we know that she is a basketball player who has a three year old child. The author seems to be saying that because this athlete had a three year old child to take care of, it was more difficult for her to perform in her sport than it is for a male. And, because the author has tied this situation to the "if women were a country" phrase, we can assume that the author means that in a country run by women, or where women's issues were taken as seriously as men's, then people would appreciate that more.
However, what is not entirely clear (again, only going by the quotes provided) is why the author is saying why it would be more difficult for a female athlete with a three year old child than a male athlete taking care of a three year old child. I'm definitely not saying myself whether or not it is or should be or anything, and maybe if one knew the specifics about Candace Parker it would be clear. I'm just saying that the gender equality issue is less immediately accessible than the example of the Japanese women's team which is clearly being given second class treatment compared to the men's team.
In any case, both examples are trying to state simply that women should be given equal opportunity, interest, and care as male athletes.
Whether or not "if women were a country" was actually a buzzword or not is something that can only be verified by looking at articles about the Olympics and what was reported as said. It's not a language issue that can be confirmed here.
lastly, it's a little difficult to say what would be meant if we said, "if men were a country." The presumption that most would agree to is that men still hold more power than women in almost all societies, and so we don't need to ask "if". If men were a country, then we would see the world look exactly the same as it looks now. So, for example, because "men are a country" in Japan, the male team rides first class, the women ride coach.