Edit: My first pass at an answer perhaps wasn't the most helpful. I've refined this to include a comment about the geometric intuition that helped me first digest the OLS.
The short answer is the Gauss-Markov theorem. The Ordinary Least Squares (OLS) regression is the Best Linear Unbiased Estimator.
Maybe a quick bit of intuition that will satisfy you is to consider how we usually calculate the distance between two points in space: using Pythagoras' theorem. Of course, Pythagoras' theorem quite explicitly takes the form of a sum of squares, and this is geometrically why the OLS estimator gives us the line which has the least geometric distance to all of the points (and this generalises to higher dimensions just as well).
It's also worth considering that your proposed estimator should really involve an absolute value somewhere. You wouldn't consider a model that substantially overpredicts half the time and substantially underpredicts half the time to be 'as good as' one that predicts as correctly as possible all the time, so you need an absolute value (a squared function implicitly takes the absolute value, since $|x|^2 = x^2$).
Given a range of different unbiased estimators, how would you decide which one is the 'best'? They all have zero bias, so you have to look to the next moment; which one has the least variance? The OLS uniquely (by the Gauss-Markov theorem) has the least variance of all unbiased linear estimators. In general we want to ascribe more weight to correcting larger errors than smaller ones, and to be precise it turns out that the square of the difference is exactly the right notion for this, but if you want the details you'd have to track through the Gauss-Markov proof.