It's an old joke but how would it actually affect the food chain and our everyday lives?
There are other threads about flying animals and characters but I haven't found one that deals with the specific issues of their effect on human society. Additionally these are super-pigs which is also different.
Pigs fly by the same method as Superman (whatever that is).
Pigs are super-beings to the same extent that Superman is. They have X-ray vision when needed and their skin will stop bullets, etc. They are not as intelligent as people so they don't use their abilities as well as Superman does. They only have normal pig intelligence.
We do not have Superman or Kryptonite to protect us. If you like there may be very limited amounts of Kryptonite available but the entire world supply is only sufficient to weaken a dozen pigs at most and pigs are everywhere.
Pigs eat more or less what they do now. They are omnivores. Also they need to eat just like normal pigs.
EDIT to be more specific about Superman's powers.
It would be impractical to post all the superpowers here. I therefore refer you to the following site. Powers and abilities of Superman If there is any conflict then I prefer powers that he had from the 1978 Christopher Reeve movies or later. As mentioned, they don't have super intelligence.