Eggipus Rex is a secret level that contains one clear Gem and a Relic and can only be accessed via Level 11: Dino Might! To get it though several steps must be taken.
Step 1:
Earn 10 Relics (any value, doesn't matter if Sapphire, Gold or Platinum) to unlock Level 27 in the Secret Warp Room. While in the center of the Warp Area (from where you select levels) stand on the orange button in the middle. It will bring you to the basement where level 27 can be started.
Step 2:
Play Level 27 'Hang' Em High' and collect the Yellow Gem at the end and complete the level afterwards to make it count.
Step 3:
Start Level 11 'Dino Might!' (Warp Room 3). Then about one minute into the level before the dinosaur egg you can activate the yellow gem path. This is why you need the Yellow Gem because without it you only see the outlines of the platform and cannot activate it.
Step 4:
The secret exit is found on this Yellow Gem path. Keep playing until the big dinosaur chases after you. At the start of the dinosaur chase there are two pterodactyls (flying dinosaurs) sitting on tree trunks. Jump onto the second one to trigger the secret exit.