Applying correct light physics to gaussian blur formulas for glow
Yes, your theory is correct. A gamma-correct blur entails converting the input pixels to linear color space, performing the blur weighting and accumulation in that space, and then converting back to gamma space at the end.

As noted in the comments, the actual transform is not literally squaring and square-rooting, that's just an approximation (and not that good of one). For the true sRGB gamma transform, see the equations in this Wikipedia article (look down the page for the equations involving $C_\text{srgb}$ and $C_\text{linear}$).

By the way, some visual comparisons of gamma-correct and gamma-incorrect blurs can be found on this page by Elle Stone, which shows why this whole thing matters.