For the eggs, bring water and vinegar to a simmer in a high-sided sauté pan. Break the eggs, one at a time, into a ramekin and carefully slide the egg into the simmering water; repeat with remaining eggs. Remove with a slotted spoon to a plate lined with paper towels and season with salt and pepper to taste.
While the eggs are cooking, cut the avocado halves into a small bowl, season with salt and pepper, and coarsely mash with a fork. Fold in the green onions.