Preheat oven to 425°F. Combine rice, water, and a generous pinch of salt in a medium saucepan. Bring to a boil. Lower heat to maintain a steady simmer, cover, and cook until grains are tender and some have burst open, 45 to 50 minutes.
Slice squash in half lengthwise, and scoop out seeds. Slice crosswise into 1/2-inch thick crescents. Toss squash and brussels sprouts with the oil, and season with salt and pepper. Arrange in a single layer on a rimmed baking sheet. Roast until tender, 20 to 25 minutes, stirring sprouts and flipping squash halfway through.
Meanwhile, add chicken and ginger to a large saucepan in a single layer, and cover with cool water by 2 inches. Bring to a boil, then reduce the heat to low, and simmer until chicken is cooked through, 10 to 12 minutes. Transfer chicken to a cutting board, and shred meat with two forks.
For sauce, place all ingredients in bowl of a food processor or blender, and process continuously until well combined. Thin with water, if desired.
To serve, divide rice among 4 bowls. Top with shredded chicken, squash, brussels sprouts, and arugula. Drizzle with sauce, and garnish with pumpkin and pomegranate seeds, if desired.