Yes, changing the direction of the normal vector to a surface would change the sign on the integral just as changing the direction the boundary is traversed changes the sign so the two sides will be equal as long as you are consistent. The way I learned it was- imaging walking around the boundary with yourightside to the interior. Then the normal vector should be point "up" with respect to you.

The ideas of "inward" and "outward" do not apply here- they only apply to a closed surface which has no boundary.