Common:
Egg - Bunny - 94,88 (avg: 91, 9.1% chance overall)
Egg - Piglet - 92,86 (avg: 89, 8.9% chance overall)
Egg - Goofy - 73,86 (avg: 79.5, 7.95% chance overall)
Egg - Lucky - 79,79 (avg: 79, 7.9% chance overall)
Egg - Teddy Bear - 83,73 (avg: 78, 7.8% chance overall)
Egg - Teddy Bear Pink - 77,75 (avg: 76, 7.6% chance overall)
Egg - Patch - 81,69 (avg: 75, 7.5% chance overall)
Egg - Pooh - 64,83 (avg: 73.5, 7.35% chance overall)
Range of the "common" group: 64-94 (30)
Overall chance of getting a "common" egg: 64.1%
Moderate:
Egg - Nala - 20,17 (avg: 18.5, 1.85% chance overall)
Egg - Pumba - 22,11 (avg: 16.5, 1.65% chance overall)
Egg - Simba - 12,14 (avg: 13, 1.3% chance overall)
Egg - Aloha Stitch - 11,14 (avg: 12.5, 1.25% chance overall)
Egg - Hatchling Baymax - 7,18 (avg: 12.5, 1.25% chance overall)
Egg - FrankenEgg - 8,15 (avg: 11.5, 1.15% chance overall)
Egg - Bambi - 12,11 (avg: 11.5, 1.15% chance overall)
Egg - Roger Rabbit - 10,13 (avg: 11.5, 1.15% chance overall)
Range of the "moderate" group: 7-22 (15)
Overall chance of getting a "moderate" egg: 10.75%
Rare:
Egg - Flounder - 4,7 (avg: 5.5, 0.55% chance overall)
Egg - Judy Hopps - 4,6 (avg: 5, 0.5% chance overall)
Egg - Orange Bird - 8,1 (avg: 4.5, 0.45% chance overall)
Egg - Fireworks - 3,5 (avg: 4, 0.4% chance overall)
Egg - Reuben - 4,4 (avg: 4, 0.4% chance overall)
Egg - Angel - 2,4 (avg: 3, 0.3% chance overall)
Egg - Bolt - 4,2 (avg: 3, 0.3% chance overall)
Egg - Stitch - 2,4 (avg: 3, 0.3% chance overall)
Egg - Tamatoa - 0,6 (avg: 3, 0.3% chance overall)
Range of the "rare" group: 0-8 (8)
Overall chance of getting a "rare" egg: 3.5%
Ultra Rare:
Egg - Honey Lemon - 3,2 (avg: 2.5, 0.25% chance overall)
Egg - Cheshire Cat - 2,0 (avg: 1, 0.1% chance overall)
Egg - Scaredy Egg Hei Hei - 2,0 (avg: 1, 0.1% chance overall)
Egg - Atlantis Crystal - 0,1 (avg: 0.5, 0.05% chance overall)
Egg - Tigger - 1,0 (avg: 0.5, 0.05% chance overall)
Range of the "ultra rare" group: 0-3 (3)
Overall chance of getting an "ultra rare" egg: 0.55%
Overall chance of getting an egg: 78.9%
DISCLAIMER: This is just from two sets of 1,000; there are assumptions made here that cannot entirely be taken as fact; for example, some put into the ultra rare category could actually turn out to be more common than these two samples show, or the Tamatoa could actually be an ultra rare, but I was just incredibly lucky.