First, least-squares methods depends on variances and covariances, and are "the best" because least-squares ensures that the covariance is the smallest.

Most people only say they square each deviation to make all the deviations positive, to avoid any information being lost due to cancelling. The other reason, which is why we don't use absolute values, is because when deviations are small (and they should be), squared deviations become even smaller.