Push the flut flut egg off the cliff was a mission that Jak and Daxter completed to obtain a power cell in The Precursor Legacy. The Birdwatcher recently saw a mother flut flut being captured by lurkers, leaving the now-orphaned flut flut egg alone high up on a cliff. She asked Jak to push it down so she can take care of the egg, promising a power cell as a reward.
After having pushed the egg down, the mission was temporarily named to Meet the Birdwatcher by the egg until Jak returned to the fallen egg and the Birdwatcher.
The PlayStation 3 and Vita versions of this mission are related to a trophy named "Eggs Over Hard", completing this mission will reward you with a bronze trophy.
From the location of the eco harvesters in "Unblock the eco harvesters", turn around and while sticking to the cliff on the left, move onto the small path leading up. This path also becomes obvious because it is here where the power cell lands after the avalanche in "Chase the seagulls". Keep moving up the path while taking care not to fall back down into the area below. The only two hazards are lurker puppies and the occasional gap in the path.
Eventually you will reach the end where a flut flut egg awaits you, from here on you will have to repeatedly hit it until it falls off the cliff. Either retrace your steps and meet the Birdwatcher by foot, or jump down for a quicker way, and talk to the Birdwatcher to receive your reward.