Heat the vegetable oil in a large pot over medium heat. Add the chicken pieces and season with salt and pepper. Cook until the chicken is lightly browned on all sides, about 5 minutes. Remove the chicken and set it aside.
Add the diced onion to the same pot. Cook for 3 minutes until it softens. Stir in the minced garlic and ginger and cook for 1 minute more.
Stir in the red curry paste and cook for 30 seconds to release its aroma. Return the chicken to the pot.
Pour in the chicken broth, coconut milk, and fish sauce. Add the cubed butternut squash. Bring the mixture to a simmer.
Reduce the heat to low, cover the pot, and cook for 20 minutes until the squash is tender and the chicken is cooked through.
Stir in the lime juice. Taste and adjust salt if needed. Ladle into bowls and top with chopped cilantro.