It would do two things:

1. It would have the same effect of making all of the values positive as the absolute value.

2. It would give bigger differences more weight than smaller differences.

For example:
2 and 4 are only 4-2=2 apart.

But, 2^2 and 4^2 are 16-4=12 apart.

Compare 2 to 12, do you see how powerful the square can be?

This is how the mean square error would be calculated:

Then you would add up the square errors and take the average.

In this case the sum of the errors is 52.1385 and the mean square error is 5.79.

Your job would be to find the line that gives you the least mean-square error. This would be the line with the best fit.

How would you do that?

You would try different equations of lines until you got one that gave the least mean-square error.

You might have to make up a table like the following as you go along:

If you get the jist of it, try the next problem.