Don't be paranoid about Bad Eggs. The whole craze started because this Australian guy bought Pokemon X and Y sometime near the release date, and then after the first or second Gym, he noticed his Pokemon turning into Bad Eggs and becoming all corrupted. His first video about it is
here, and then he made a follow-up video that explains more about where they came from
here. Basically, in the end, he figured that his personal 3DS had a bad SD card that caused the glitch to happen, because he got the exact same glitch on both his X and Y version. The chances of two of his games being corrupted are almost zero unless the problem was hardware related and not software related. They cannot be traded in any way between games, and they cannot be put on Wonder Trade. They also never hatch.
Basically, these Eggs just appear in your box for seemingly no reason, but it's likely a hardware glitch on very very few 3DSes. There have been no known instances of people getting Wonder Traded Pokemon that turn into Bad Eggs that I can find anywhere. Basically people freaked out over what isn't a problem.
However, if you do happen to get a Bad Egg somehow, the only way to get rid of it is to restart your game. But if you get the Bad Egg, it probably means that your SD card is corrupted, so you'll just get it again. Anyway, you won't lose your Pokemon if you have PokeBank. Just put everything you want to keep on PokeBank, restart your game, and the glitch is gone. The only inconvenience is you'll have to play through the story again.