In principle, this technique might work. However, you would need to consider several factors:
Does the density of new ostrich eggs vary significantly from 38 g/cm³? Does the rate at which ostrich egg density declines depend on how the egg was stored? Does this rate depends nn variability in the egg itself?

Essentially, you would need to find the error associated with egg-density dating, likely by constructing a calibration curve (year vs. density) of eggs stored in a similar manner with known density and dates. The 'width' of this plot would tell you whether or not your method is accurate, and you could find a standard deviation to give you an expected error (and "proof" - actually a confidence interval - that your egg is a certain age).

I would guess that natural variations in eggs and the storage method would render your technique less accurate than other methods for dating organic samples: see "amino acid dating" or "radiocarbon dating".

Depends on why the egg gets lighter with time. It probably is the rate of decomposition of the protein network that the egg is partially made out of, and that will depend on the temperature and humidity of the environment in which the egg is decaying: the colder and drier, the slower it will happen. So you could use this to find the egg's age, but you would have to have this caveat.

The density of water is only 1 GM/cm³. Proteins are ca 1.3 GM/cm³. Fats are not as dense as water. So the ostrich egg would have a density in this range. Where did you get your data about changes in density with time?