You can only participate in the Egg Hunt after unlocking pets.
Egg Hunt takes place at Easter. For a specific period of time, you get three tasks from the bartender in the tavern every day which you need to complete.
You'll get 25 or 50 points for every task. The goal is to collect a mininum of 350 points at the end of the event to gain a rare pet (you need to unlock it in the habitat first). If you already collected all unlocked rare pets you'll get a fruit basket with 10 fruit of each type.
On the way to the goal there are three intermediate goals: a common pet for 150 points (needs to be unlocked in the habitat). If you already collected all unlocked common pets you'll get a fruit basket with 5 fruit of each type. For 225 and 300 points collected you also get a fruit basket as a reward.
In total, more than the 350 points needed for the final reward can be collected, so you do not have to complete all tasks.