Bring a large pot of lightly salted water to a boil. Cook lasagna noodles in the boiling water, stirring occasionally, until tender yet firm to the bite, about 8 minutes. Drain and set aside to cool.
Combine eggplant, squash, zucchini, mushroom, and marinara sauce in a medium saucepan. Bring to a boil over medium heat. Reduce to a simmer and cook, covered, until vegetables are tender, about 10 minutes.
Combine ricotta cheese, mozzarella cheese, Parmesan cheese, garlic, egg, oregano, salt, and pepper in a large bowl.
Spread 1/3 the sauce mixture into the bottom of a 8-inch square baking dish. Top with 2 cooked lasagna noodles. Spread 1/2 the cheese mixture on top and cover with 2 more lasagna noodles. Repeat, ending with sauce on top. Cover with aluminum foil.
Bake in the preheated oven until golden and bubbling, about 40 minutes.