In addition to the points made by Peter Flom and Lucas, a reason for minimizing the sum of squared residuals is the [Gauss-Markov Theorem][1]. This says that if the assumptions of classical linear regression are met, then the ordinary least squares estimator is more efficient than any other linear unbiased estimator. 'More efficient' implies that the variances of the estimated coefficients are lower; in other words, the estimated coefficients are more precise. The theorem holds even if the residuals do not have a normal or Gaussian distribution.
However, the theorem is not relevant to the specific comparison between minimizing the sum of absolute values and minimizing the sum of squares since the former is not a linear estimator. See this [table contrasting their properties][2], showing advantages of least squares as stability in response to small changes in data, and always having a single solution. [1]: http://en.wikipedia.org/wiki/Gauss%E2%80%93Markov_theorem [2]: http://en.wikipedia.org/wiki/Least_absolute_deviations#Contrasting_Least_Squares_with_Least_Absolute_Deviations