For the vegetables: Warm the olive oil in a medium skillet over medium-high heat. Add the onions and cook until starting to soften, 2 to 3 minutes. Add the squash and zucchini, sprinkle with salt and pepper and stir and cook for 3 to 4 minutes. Remove from the heat and set aside.
Grill the tomato slices on a grill pan or simply sear them in a hot skillet. Remove and set aside.
For the hollandaise sauce: In a small saucepan, melt the butter until sizzling.
Put the egg yolks in a blender. Turn the blender on low to allow the yolks to combine, and then begin pouring the very hot butter in a thin stream into the blender. The blender should remain on the whole time, and you should be careful to pour in the butter very slowly. Keep pouring the butter until it's all gone, then immediately begin pouring the lemon juice into the blender. Check the blender to make sure the sauce is still liquid and moving easily through the blades; if it's not, add a little more juice and give it a stir, then blend again. Add a generous shake of cayenne pepper. If the sauce is too thick, continue to blend while adding more lemon juice.
For the poached eggs: Bring a medium saucepan of water to a gentle boil. Add the vinegar. Use a wooden spoon to carefully stir the water into a circular "whirlpool." Crack the eggs into the water and allow them to swirl around in the water until they begin to set. Let them stay in the water for 1 minute, then remove them with a slotted spoon and set on a plate. If the eggs need to be cooked longer, just add them back to the water for 30 seconds at a time.
Spoon the vegetables on plates or in bowls and set the eggs on top. Lay the tomato slices on the side. Place the cheese slices on the side. Sprinkle the eggs with salt and pepper and serve with the hollandaise sauce poured over the top.