Bring a large pot of salted water to a boil over high heat. Drop the pasta into the boiling water and cook to al dente according to package directions.
While the pasta is cooking, place a medium pot over medium heat with the olive oil. Add the bacon to the pot and cook, stirring occasionally, until golden brown and crispy, 5 to 6 minutes. Remove the bacon from the pot and reserve on a paper-towel-lined plate. Pour the residual grease out of the pot, reserving about 1 tablespoon.
Return the pot to medium heat and add the garlic; cook until aromatic but not golden, about 30 seconds. Add the flour to the pot and continue cooking for 1 minute. Add the milk and mustard to the pot and bring the liquid up to a bubble; cook, whisking frequently, until thickened, about 1 minute. Add the squash to the pot and return to a bubble. Remove the pot from the heat, whisk in the cheese, and season the sauce with salt and pepper; reserve warm.
When the pasta has 3 minutes of cooking time left, drop the broccoli into the boiling water. Drain the cooked pasta and broccoli and combine with the cheese sauce. Garnish with the reserved crispy bacon and serve.