The pigs hadn't always eaten eggs. Several thousand years ago, when humans had still roamed the earth, they had mainly lived on farms and had spent their lives eating slop and byproducts of human food production. They were content with their lives, for the most part. The food may have tasted foul to the humans, but the pigs found it satisfying and somewhat pleasant. Then nuclear war came. The humans were wiped out completely, along with the majority of the pig population. Those pigs that remained eventually evolved into little more than bouncing heads with stomachs. The pigs lived fairly peacefully for over four-thousand years, living mainly on the vegetation of the North American Midwest, which was one of the few places on earth that could still support even the highly mutated pigs and birds.
Unfortunately, this peace could not last. After four-thousand years, radiation on the surface had begun to clear up, but millennia of weathering and erosion had begun to take its toll on the land. Eventually, a particularly bad tornado had been able to tear away the last shreds of protection around an ancient nuclear waste dump. The toxic waste quickly made its way into the water. The pigs didn't need much water anyway, they could get what they needed from the well-hydrated plant life, but food quickly became a problem. Plant life, after absorbing the polluted water quickly became toxic and the pigs had to begin searching for a new source of food. They soon discovered the bird eggs.
Evolution had served the birds well. Unlike the pigs, who had simply evolved to survive in high levels of radiation, the birds had evolved to thrive. One species developed extreme strength, and another extreme speed. Other species had gone further. One had three complete sets of organs and could divide into three organisms at will. One could blow itself up if need be and another more terrifying species could both lay eggs at will and use telekinesis to blow up its own young.
Along with these bizarre adaptations, the birds had enzymes that not only allowed them to survive in areas of high toxicity, but the use the toxic waste to their advantage. These enzymes were especially prominent in the fluid within their eggs, which the pigs would eventually discover and exploit.
It was actually fairly easy for the pigs to retrieve the eggs. Despite their many adaptations, the birds' senses were actually quite dull. The problem lay in keeping the eggs. The birds would stop at nothing to retrieve the eggs, and even once the pigs had developed complex architectural structures to protect themselves, the birds developed catapults which allowed them to launch powerful attacks. The weapons combined with their superior skills made them formidable opponents.
Battles between the birds and the pigs went on for several centuries until the pigs developed the enzyme too and realized that they could survive without the eggs. Of course, nothing can last forever, and eventually, both the birds and the pigs died out, making way for new species.