First it is not returns but residuals and squared residuals of your model that should be IID.
Second, it is required that squared residuals are IID because you are not only interested on your mean forecast but also about the variance of your forecast.
Imagine two cases :
- your forecasted value is 0.5 but its variance is unknown because residuals of your models have time-varying variance. The variance of your forecast may be 0.2 or 0.8, you simply don't know it. (in fact you only know its central tendency)
- your forecasted value is still 0.5 but you know for sure that it has a certain variance (let's say 0.2) because it is derived of the variance of your squared residuals that is constant.
In the second case you are able to build a confidence interval about your forecast but not in the first case.
To sum up, residuals must be IID to be sure that the mean forecast is correct. Squared residuals must be IID to infer the variance of your forecast and to be able to build multi-step forecast and confidence bands.