There are only 7 available sums with those numbers (-3,-2,-1,0,1 2, and 3)

Consider the sums of 3 and -3.

It is not possible to have a row that sums to 3 (all 1's) and a column that sums to -3 (all -1's).

And if two rows sum to 3 and -3, then the column sums can only be different if the 3rd row is -1 + 0 + 1 = 0.  But then the third column sums to 0, same as the third row.

So, we cannot have both a sum of 3 and a sum or -3.

Without loss of generality, assume that we do not have a sum of -3.

Then the 6 sums in the grid are  (-2,-1,0,1 2, and 3).  

But this sums to 3, which is an odd number.

This is impossible, because the sum of the rows equals the sum of the columns, so the total sum of all rows and columns must be even.

q.e.d.