Given that the word is constructed out of the Japanese language, it would be an understandable mistake to think the word was Japanese.
However, "flyjin" is not a Japanese word.
The word "flyjin" was coined within the English speaking foreign community in Japan, where the issue of people leaving Japan after the earthquake was a topic of much discussion.
Japanese people, for the most part, are unaware, and unconcerned, about the issues that gave rise to the term. Thus the term "flyjin" is almost entirely unknown to Japanese people.
Following from that, there is no widely accepted "correct" way to write it in Japanese. Nor is there an accepted Japanese equivalent term. If you are speaking to a Japanese person about "flyjin", you will have to explain the concept however you see fit.
For antonyms, they also exist only within the subculture of English speaking foreigners in Japan, and would not be considered "Japanese" words. Still, there are two contenders. One is "stayjin", the meaning and origins of which are self evident.
Another is a term coined by a friend of mine: "fryjin". It refers to "fry", as in those who stayed after the earthquake were "fried" by the radiation from the Fukushima nuclear power plants.
The best part about it is that in katakana, it's the same as "flyjin": フライジン. So Japanese can't tell the difference between the words, just as they don't really differentiate between foreigners who stayed and those who didn't. I like the metacontextual implications.