Mix ricotta, mozzarella, and chives; season with salt and pepper. Transfer to a piping bag or resealable plastic bag (cut tip). Pipe into squash blossoms. Place breadcrumbs in a baking dish. Lightly beat eggs in another dish. Heat oil in a large skillet over medium-high heat. Dip blossoms in egg, then breadcrumbs. Cook, turning once, until golden, about 4 minutes. Drain on paper towels; season with salt.