Heat the oil in a large, heavy pot over medium-high heat. If you like a lot of heat, stir in 4 ounces of the curry paste, and if you'd like it milder, use 2 tablespoons of the paste. Cook the curry paste in the oil, stirring frequently, for 2 minutes. Scoop the thick layer of coconut milk solids out of the cans, leaving the watery milk, and stir into the pot. Stir for 2 minutes, then add the chicken pieces. Sauté for 1 minute. Stir in the remaining coconut milk. Reduce heat to medium-low, and simmer, stirring occasionally, for about 10 minutes, until the chicken is cooked through. Add the squash, and simmer for 4-5 minutes, until just tender.

Taste, and adjust the seasoning with fish sauce, sugar, and soy sauce. Serve hot, with rice or rice noodles.