The reasoning is really pretty simple. First we'll examine why a (solid) Dyson sphere is not dynamically unstable; then we'll apply the same reasoning to determine why a Niven ring is.

Take a uniform spherical shell, and a test particle in that shell somewhere, but not in the center (the center is a trivial case). Consider the gravitational attraction from two sides of that shell based on a double-cone, with its apex on our test point. Let's take into account only the gravitation from the portions of the sphere which this double-cone intercepts. Since we're nearer to one side of the sphere than the other, the nappe of the double-cone which intersects the near side will be closer but will intercept less of the sphere; and the nappe on the far side will be further away but will intercept more. Take r to be the distance from the test point to one of the nappes. Since we're inside a sphere, the portion of the sphere that will be intercepted will be proportional to r². Gravity, of course, varies as 1/r². These two effects cancel out, and what you end up with is the (somewhat surprising) conclusion that the gravitational attraction of a portion of the sphere is independent of your distance from it. As the original poster said, the nearer side is closer, but there's more of the further one. Since we're talking about two nappes here, these two distance-independent forces are equal and opposite, and so you have zero net force.

Conclusion: Since the choice of position of the test particle was arbitrary and the orientation and solid angle of the double-cone was arbitrary, we have found (non-rigorously, of course) Newton's first theorem: The gravitational attraction inside a uniform spherical shell is zero.

Now let's see why this doesn't apply to rings. Again we have the same kind of situation, with an off-center test particle (again, because the centered case is trivial) and a double-cone, although now since we're effectively dealing in two dimensions, it's more like two sectors of a circle. Now the area intersected by each sector is only proportional to r, while gravity is still proportional to 1/r². Now you get a situation where the gravitational attraction of each portion is not independent of distance; it varies as 1/r. That is, the closer side will attract you more strongly than the further side. Put a test particle off-center in the ring, and it will crash itself against the ring. Because of Newton's third law, put an object more massive than the ring itself off-center in there and the ring will smash itself against it.

Thus, Niven rings are dynamically unstable.