What happens to Variance when every observation is squared? What is the effect on the Variance of a set of observations if every observation in the set is squared?

Well, variance is independent of translations, but squaring is not. That is, if you add the same constant term to each observation, you don't change the variance, but you will change the square of the observations, and you will also change the variance of the squares. So this is a not-entirely-rigorous argument showing that the effect on variance of squaring the observations cannot be easily characterized without knowing exactly what the observations are.
Note that as variance is independent of translation, it's also independent of zero-point. So if one person measures temperature in Celsius and another person measures them in Kelvin, they will get the same variance. But if each person squares their numbers, they will get very different data sets with different variances.
