The female flowers of an acorn squash plant can be pollinated with pollen from another plant in the squash family. If the hybrid seed is planted the resulting fruit will not have the same characteristics as the original acorn squash parent and may be of poor quality. If this was the cause of the bitterness in the fruit you picked then all of the fruit from the plant would have this characteristic.
Bitterness can also occur on certain fruit from a squash plant if it was stressed by lack of water or some other condition. To avoid this problem only plant the seed of a known variety and provide good growing conditions, especially adequate fertility and irrigation. The roots of squash can spread a few feet from the base of a plant. A large area of soil around each plant should be watered to wet the plant’s entire root zone.
By V. Lazaneo, Urban Horticulture Advisor, Emeritus, UC Cooperative Extension, August 2012