In a medium saucepan, bring the water and salt to a boil over medium-high heat. Add the squash and cook until tender, about 10 minutes. Strain and set aside.
For the dip: In a food processor, blend the bell pepper, avocado, ground cumin, yogurt, and mayonnaise until smooth, about 1 minute. Season with salt and pepper, to taste.
Transfer to a serving bowl and let stand for 10 minutes to thicken.
Serve the dip with warm or room-temperature chayote squash.