i researched thermal capacity last time it came up, and per unit of volume water bottles have the highest thermal capacity. higher than water with salt or otherwise impure water, higher than stone or ceramics, even higher than blocks of solid metal. by volume, water beats a block of copper even while the copper is around 7-8 times heavier by volume. so as stabilizers, sealed PET water bottles are totally the way to go.

if the individual egg boxes are completely sealed, i see no reason to place the temp probe inside an egg box, since there should be perfect or near-perfect thermodynamic equilibrium. (can be confirmed with temp gun). if the egg boxes are not completely sealed, there may be evaporation cooling going on, so a probe inside an egg box makes sense. again, a temp gun and test runs will clarify things.