A system is in equilibrium when there is zero net force (all the forces balance), but to be stable more is required – when a stable system is disturbed slightly, it tends to return to the equilibrium, while an unstable system does not. For example, a pencil balanced on its point is in equilibrium, but is not stable. A ball in a bowl is in equilibrium and stable.
This chart shows different kinds of stability. The lines are potential energy functions. The oscillating Ringworld is as stable in the vertical direction as a ball in a bowl is. Any perturbation simply causes an oscillation or changes the magnitude of the oscillation – and if there’s any friction in the system, the effect of the perturbation eventually dissipates leaving the Ring centered.
In the horizontal direction though, the Ringworld is completely unstable – the slightest perturbation causes a complete disaster as the Ring tends to get more and more off center.
The Dyson sphere is neutrally stable. Any perturbation just moves the Sphere slightly but gives it no tendency to move further along or to return to the original position – like a ball on a table.
Finally, attitude jets would make the Ringworld is meta-stable - it will recover from small perturbations, but be vulnerable to a large perturbation. Most systems in the real world are metastable; they will stay put, if they’re not pushed too far - just like most people.
It’s worth noting that if the impact of light on the shadow squares causes a greater outward force than gravity produces an inward one, then the shadow squares are stable horizontally but unstable vertically. Because the dominant force is outward, not inward, all the signs reverse making stable situations for the Ringworld into unstable ones for the shadow squares and vice versa.