In a pot of boiling, salted water, cook the pasta according to package directions. Drain and set aside.
In a medium saucepan, heat the olive oil over medium-low heat. Add garlic and cook, stirring frequently, until the garlic is golden, 1 to 2 minutes.
Whisk in the flour until smooth, about 2 minutes. Gradually whisk in the chicken broth, Greek yogurt, milk, pepper, and oregano. Bring to a low boil, stirring constantly. Lower the heat and simmer, stirring gently, until the mixture thickens, about 3 minutes. Stir in 3/4 cup Parmesan.
Add cooked chicken, broccoli, and squash to sauce mixture, followed by the cooked pasta. Toss to combine and serve with more Parmesan, if desired.