Synchronize doesn't work on eggs, both ones that you breed for yourself and gift eggs from NPCs.
However, from Pokemon Emerald onwards, the Everstone can have a chance to pass down the nature of the parent holding the Everstone. This differs a bit in the earlier gens, with female parents and Ditto having a 50% chance to do so in Emerald and DPPT, either parent having a 50% chance in HGSS and BW, 100% chance to pass a nature from B2W2 onwards, and apparently it doesn't work with the Masuda Method in Gen 4.
Source: This, this, and Experience, I've bred Pokemon with Synchronize to get different natures before.
Hope I helped!
