Have you considered a Monte Carlo simulation or Bootstrap?
For Monte Carlo, simulate a random number for each value, using the distribution of your error bounds (or any distribution you feel appropriate) and then compute your Gini index. Repeat many times and get a distribution of the estimated Gini index. To create a confidence interval, consider using 1.96 * SE intervals. Also consider taking 2.5% and 97.5% percentiles.
For Bootstrap, it sounds like each value has a bunch of data behind it. So for each value, sample with replacement from the value's data to create a new sample with the same size as the original. Recompute each value using their own new sample and then compute the Gini index. Repeat the entire process many times to create a distribution of the estimated Gini index. There are many ways to create a Bootstrap confidence interval, but taking 2.5% and 97.5% percentiles is easy and common.
If you try both methods, I'd be curious how they compare.