I don't know where you heard that but it is not the case as many of us have seen. While a decrease in the length of daylight can trigger a molt even that waits until there is quite a bit of difference in the fall. If you look at the change in length of daylight around the Summer Solstice it's barely noticeable.
Before they were domesticated chickens would lay eggs, hatch chicks, and raise them during the good weather months when food is plentiful. When the weather and food supply turned bad in the fall, they would replace worn out feathers, stop laying eggs, and rest. In the spring when the weather and food supply got better they'd go back to laying eggs and hatching chicks. They were not going to stop brooding just because of the position of the sun, the best days of summer were still ahead of them.
When we domesticated them our selective breeding changed them. They no longer all go broody as soon as they got a clutch of eggs. Some never go broody at all. We saw that they had good food year around, not just in summer. They are around artificial light which sometimes messes up the days getting longer or shorter, either on purpose or by accident. Some people keep them warmer in winter.
While most chickens still keep the basic schedule and molt in the fall, some hens will start laying again right after they finish the molt, even if it is before the Winnter Solstice. While most hens that go broody wait until the warmer days of spring and summer, some will go broody in the middle of winter.
I don't know if any of your hens will go broody this summer, but you don't have to worry about June 21st being a cut-off date. Probably half of my broody hens hatch after that date.
