No, can't say I've ever heard of that old farmer's saying. However, it certainly resonates with some old farmer sayings that are often validated to some extent by science. However, I’m not sure this old farmer’s saying holds much weight.
Certain plant genera and plant families are physiologically predisposed to maintaining a portion of their post nutrient translocated and desiccated foliage throughout the dormant season (i.e., marcescence). Many oak species (genus: Quercus), especially juvenile oak trees, will maintain a certain percentage of their fall leaves into the start of the next growing season. American beech (genus: Fagus) falls into the category too as oak and beech are in the same plant family (Fagaceae). Other species, such as eastern hophornbeam (i.e., ironwood) and American hornbeam (i.e., blue-beech or musclewood) also maintain a portion of their foliage throughout the dormant season. Granted not all the aforementioned species exhibit this phenomenon with 100% regularity, though it is certainly a very common occurrence—especially in juvenile trees. Ultimately, in a physiological context, the abscission layer (separation zone at the base of the petiole, or leaf stalk) fails to abscise or separate upon autumn leaf senescence. This in turn causes complete leaf senescence to be delayed until spring. Again, certain genera or plant families are physiologically predisposed to this phenomenon, otherwise known as marcescence (i.e., retention of dormant, desiccated plant organs). Environmental conditions and genetics presumably play a role in marcescence, although I’m uncertain as to what extent.
Thanks for the trivia!