Bring a large pot of salted water to a boil. Add the spaghetti and cook according to package directions for al dente.
While the pasta cooks, trim the top and bottom off of each squash. Peel the squash in ribbons into a colander, using a vegetable peeler, turning the squash as you peel. Stop peeling the squash when you only have the center core of seeds left. Discard the core and seeds.
Reserve 1/4 cup of the pasta water, and then drain the pasta over the squash ribbons. Put in a medium bowl and toss to evenly distribute the pasta with the squash ribbons.
Toss the shrimp with 1/2 teaspoon kosher salt and some pepper. Heat 1 tablespoon oil in a large skillet over medium-high heat. Add the shrimp, spread into a single layer and cook without stirring, until just turning pink around the edges, about 2 minutes. Stir the shrimp, add the tomatoes, garlic and pepper flakes and continue to cook until the tomatoes have softened and the garlic has toasted, about 1 minute. Add the pasta and squash ribbons, broth and the remaining 1 tablespoon olive oil. Cook, tossing, until warmed through and most of the liquid has been absorbed. Add the pasta water a bit at a time, if needed, if the noodles seem dry.
Season with additional salt and pepper. Divide among 4 bowls and top with the chives.
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