If you want to get the Royal Egg of the Bearers, you will need to play the game Bear. This is a fun game where you will be playing either a bear or a survivor. Bears need to hunt down the survivors in a maze-like level, while the survivors need to make it to the end of the round.
No matter which role you end up playing, you need to try and find one golden key. There will be multiple keys hidden randomly around the levels, so just keep playing until you manage to track one down. When you find the key, you can finish the round, then make your way to a brown door in the main lobby.
The door will have been locked when you first joined the lobby but will be open now that you have a key. Head inside, and you will be transported to a strange art gallery. There is a note on the floor that will contain clues.
Each clue references one of the pictures on the wall, and each picture has a letter on it that signifies a color. The order of the clues, and the subsequent colors, is a code that you will need to enter into a panel in the next room.
The clues, and colors, are random, so it is hard to offer any help on the solution that you need. Once you figure out the order of colors, you need to go to the next room and enter them into the wall panel in the correct order, then hit the green button.
A secret door will open, which will teleport you to an obstacle course. Make it through the obstacles to get to a room that contains the egg. Head inside, then hit the red button and grab the egg. Congratulations, as you now have the Royal Egg of the Bearers.
As a note, both doors were pretty slow to transport us to the next segment, so you might need to wait a little while for the game to realize what it needs to do.