In My Experience, when using an older egg, this is exactly the problem-the egg contents start to seep out.
Sweating
Egg shells are porous, especially goose eggs. Over time the porous shells have let in enough air to float. I call it "sweating", the egg contents start to sweat, from lots of points/pores, wrecking the dyed parts. I have had this happen to me on goose eggs that I left much too long in the refrigerator bin. Mark the Egg Date
It is a good idea to write in pencil on the base of a goose egg, for example, a month and date: 5-14; so you know how old the egg is, if you don't plan to use the eggs immediately, while they are fresh and being stored.
It's Not Worth It using an old egg, or an non-refrigerated egg because all the hard work gets wrecked. I have salvaged a sweating egg, by promptly draining it, and rinsing the inside well. Then I carried on finishing the problem egg, with not much of a disturbance to the pattern.

Egg shells are porous, especially goose eggs. Over time the porous shells have let in enough air to float. Place the suspicious eggs in water to check their freshness. Most "stale eggs" will not "sweat". It is the floaters-"very old" eggs that can cause problems during your Pysanky writing, and may cause spotting of the dyed areas.

- Using a syringe, fill the egg with a mixture of baking soda and water overnight. Flush out the contents and repeat if necessary.
- Do not use vinegar, as it can etch thru the shell.
- I also store the shells with a dryer sheet, but not touching the eggs.