In a large cast-iron skillet, heat 2 tablespoons olive oil over medium-high heat. Add the arugula to the skillet, season with salt and pepper, and cook for approximately 4 minutes or until just wilted, stirring. Transfer to a warm bowl.
Add the squash to the skillet, season with salt and pepper, and cook for 2 minutes, stirring occasionally. Add the lemon juice and the honey to the skillet. Continue to cook for 5 minutes longer, stirring. Remove from the heat. Add the squash to the bowl with the arugula and toss to combine. Set aside.
Poach the quail eggs.
Transfer the arugula and squash to a serving dish. Arrange the baguette rounds on the serving dish. Put the quail eggs on top of the arugula. Sprinkle with black pepper, if desired, and drizzle a little olive oil over the top. Serve immediately.