There are $9^{7}$ total shapes possible, since each of the 7 columns can contain anywhere from 0 to 8 checkers. The number of shapes symmetric with respect to a horizontal flip is the number of shapes of the leftmost four columns, since the configuration of these four columns uniquely determines the configuration of the remaining columns if it is known the shape is symmetric: $9^{4}$. Now we know there are $9^{7}-9^{4}$ non-symmetric shapes, so there are $\frac{9^{7}-9^{4}}{2}$ non-symmetric shapes modulo flips. Thus the total number of shapes modulo flips is $n=3^{6}=729$.