What causes seasons on earth? The change in length of day and angle of incidence of sunlight as the earth revolves around the sun changes the amount of heat the sun provides to a particular place on the earth. The distance from the sun to the earth does change but plays no significant effect on seasons - in fact the Earth is closest to the sun during January (northern hemisphere's winter).
On Ringworld, the length of day is constant. While this could be changed by adjusting the distance between the shadow squares (note that the shadow squares aren't in orbit around the sun either - they're connected by improbably strong wires and are spinning faster than necessary to stay in orbit), it's hard to imagine getting a useful amount of day length change over a long period and the mechanism to do so would be pretty complicated. Changing the distance to the sun is also possible (but see what happens two slides further), but since the Ringworld rotates with a period of 9 days or so, seasons would be very short.
But it turns out that there is one other thing to change about Ringworld.