Starting at the stem end, run a vegetable peeler down the length of each squash, using very little pressure, to create long, thin ribbons. Continue making ribbons until you reach the seeds in the middle of the squash. Turn the squash over and repeat.
Arrange half the ribbons in a single layer on a rimmed baking sheet. Drizzle the squash with half the lemon juice, 1/4 cup of the oil, 1/2 teaspoon of the salt, and 1/8 teaspoon of the pepper. Scatter half the garlic slices and 12 of the basil leaves evenly over the squash. Cover with the remaining ribbons in a single layer, then add the remaining lemon juice, oil, salt, pepper, garlic, and basil.
Set aside to marinate until the ribbons become translucent, about 1 hour. To serve, use a fork to twirl each ribbon into a loose bunch and place in a bowl.