For the poached eggs:
Pour enough water into large roasting pan to reach a depth of 2 inches. Pour enough water into a large skillet to reach a depth of 1-½ inches; add 1 teaspoon salt to skillet and bring to a simmer.
Crack each egg into a separate custard cup. Working with 4 eggs per batch, gently slide 1 egg at a time into simmering water in skillet. Cook just until egg whites are set, about 1 minute (yolks will be only partially cooked).
Using a slotted spoon, carefully transfer eggs to prepared roasting pan with water. Reserve skillet with water.
For the hash:
Chop leftover vegetables roughly and place in a large bowl. Add eggs and corned beef. Season to taste with salt and pepper; stir well. Set aside.
In a large frying pan over medium-high heat, add oil and sauté onion until softened, about 5 minutes.
Add corned beef hash mixture and cook until golden brown on bottom, occasionally pressing down with spatula, about 4 minutes. Turn hash over in small portions and cook until second side is golden brown, occasionally pressing down with spatula, about 3 more minutes.
Meanwhile, bring water in reserved skillet (from poached-egg preparation, top) to a simmer. Using a slotted spoon, gently transfer poached eggs from roasting pan back to skillet. Cook just until yolks are softly set, about 2 minutes.
Divide hash among plates. Top with one or two poached eggs and serve. Season to taste with salt and pepper.