Bring a large pot of water to a boil. Salt the water, add the fettuccine and cook until al dente. Reserve 1/2 cup of the pasta water, then drain the pasta.
Using a vegetable peeler, shave the squash into ribbons. In a large nonstick skillet, heat 2 tbsp. oil over medium-high. Add the squash; cook, tossing occasionally, until tender, 3 to 4 minutes. Transfer to a bowl.
Add the remaining 1 tbsp. oil to the skillet; heat over medium-high. Add the onion and garlic; cook, stirring occasionally, until starting to soften, about 3 minutes. Add the tomato and capers; cook until the tomato softens, about 2 minutes. Add the squash; cook until heated through, about 1 minute. Add the fettuccine and 3 tbsp. pasta water. Cook, tossing often and adding more pasta water by the tablespoon if dry, until the pasta is heated through, about 1 minute. Off heat, stir in the feta and mint. Season with salt and pepper.