When you have either a 'Banjo-Kazooie: Nuts and Bolts' or a 'Banjo-Tooie' gamesave on your hard drive the following items will appear at the specified location and will become unlocked when you complete the corresponding task.
Unlock Red Egg: Located in the cabins quarters in Rusty Bucket Bay.
Unlock Ice Key: Go in Wozza's Cave in Freezy Peak. There will be a ledge on the right you can backflip onto. The key will be through there.
Unlock Blue Egg: Located in Gobi Valley. You must first fly to the ledge on the left and go through the door. Slam on the switch and the coffin will open, egg inside.
Unlock Pink Egg: Go to Treasure Trove Cove and Shark Food Isle will be raised. Inside is a giant slope you have to hop on to get to the top where the pink key is.
Unlock Turquoise Egg: Located in the wine cellar of Mad Monster Mansion. A formerly locked wine barrel will be open and lead to the blue egg
Unlock Yellow Egg: Located in Click Clock Woods in winter. Located in Nabnut's House.
Unlock Green Egg: Located on the toilet room of Mad Monster Mansion, on top of the toilet itself.