Nope. Load following is something different from a planned shutdown. You can order your reactor to support load following, most reactors in Europe are built with that capability. This means that for a boiling water reactor, the reactor will regulate itself due to feedback from the cooling system: Slower coolant flow due to less load on the generators will increase the temperature in the reactor core, creating more steam bubbles that slow down the reaction and therefore reduce power output. Pressurized water reactors (PWRs) can also be configured for load following, however, they are slower and need to employ control rods.

The usual figures given for PWRs are something like "These reactors have the capability to regularly vary their output between 30–100% of rated power, to maneuver power up or down by 2–5%/minute" (https://en.wikipedia.org/wiki/Load_following_power_plant#Nuc...). This means that you can regulate your PWR over its power range within slightly more than half an hour at worst.

There is a lower limit to how much you can regulate your reactor down due to various limitations like Xenon poisoning. Because of that, a controlled complete shutdown is slower, because you need to get through the "icky" region between e.g. 30% and 0%. Also, you usually prepare for the longer phase the reactor will stay shut down. But all in all, shutdown is different from load following, and a conclusion that nuclear plants can't load-follow from a question about shutdown is just asking the wrong questions and drawing questionable conclusions from it.