Put the cubed squash in a pot with 2 cups of the water. Bring to a boil, then simmer, covered, until completely fork-tender, about 15 minutes.
While the squash cooks, make the dumplings: mix the glutinous rice flour with the warm water until it forms a soft, workable dough, adding a few more drops of water if it's too dry to hold together. Roll into marble-sized balls — you should get 16 to 20.
Separately, cook the rinsed rice in the remaining 2 cups water until it's fully soft and starting to break down, about 15 minutes. This becomes the base that thickens the whole porridge, so don't rush it.
Blend the cooked squash (with its cooking liquid) until smooth. A stick blender in the pot is easiest; a stand blender works in batches.
Combine the blended squash and the cooked rice in one pot. Stir well — you can blend this together too for a completely smooth porridge, or leave the rice slightly textured if you like more bite.
Bring the combined mixture to a gentle simmer over medium-low heat, stirring often so it doesn't catch on the bottom. It will be quite thick — thin it with a splash of water if it's closer to paste than porridge.
Drop in the rice dumplings and simmer for 4 to 5 minutes, until they float and turn slightly translucent.
Stir in sugar and salt, tasting as you go. Simmer another 2 minutes to let the sweetness settle in, then serve hot.