Cover potatoes with cold salted water by 1 inch in a large saucepan.
Bring to a boil, then reduce heat and simmer until potatoes are just tender, about 20 minutes.
Drain and cool, then cut into bite-size pieces.
Heat oil and 1 tablespoon butter in a well-seasoned 10-inch cast-iron skillet over high heat until foam subsides, then sauté potatoes with salt and pepper to taste, turning once or twice, until nicely crusted, 8 to 10 minutes.
Transfer to a serving bowl.
Sauté corn and squash in remaining 3 tablespoons butter in skillet over moderately high heat, stirring, until crisp-tender, about 5 minutes.
Stir in beans and sauté, stirring, until heated through.
Season with salt and pepper and add to potatoes with onion and chives, stirring to combine.