Put the quinoa in a medium pot and cover with cold water; let soak 5 minutes. Drain thoroughly in a fine-mesh strainer and return to the pot. Pour in 1 1/4 cups water and bring to a boil over high heat. Cover and reduce the heat to low, then simmer 15 minutes. Remove from the heat without disturbing the lid and allow it to rest for 5 minutes. Fluff with a wooden spoon or fork.
Arrange the quinoa, salad mix, strawberries and chicken in lines in a large, wide shallow dish or in 4 small shallow dishes. Serve oil, vinegar and salt and pepper on the side.