Transfer rice to a large baking dish or bowl. Slightly cool. Sprinkle with rice vinegar; toss until combined. Cool to room temperature. Cover with damp towel to prevent rice from drying out.
Meanwhile, for spinach-egg filling, squeeze excess moisture from spinach. Finely chop spinach. Combine spinach, eggs, and salt in small bowl. Heat oil in medium skillet over medium heat. Add spinach-egg mixture. Cook, stirring occasionally, until cooked through but still glossy. Remove from heat. Cool to room temperature.
To assemble sushi rolls: For each roll, place seaweed sheet, shiny side down, on bamboo matt. Using wet hands, spread a scant 1 1/4 cups sushi rice evenly on seaweed sheet to within 3/4 inch of edge nearest you and opposite side. Place 1 1/2 or 2 slices of ham across the center of the rice. Spoon about 3 tablespoons of spinach-egg mixture in a strip across the ham.
Use bamboo matt to lift and tightly roll sushi, jelly-roll style. Shape roll by lightly pressing roll together with damp hands. If desired, tightly wrap sushi rolls in plastic wrap and refrigerate for up to 2 hours. To serve, using a wet knife, slice each roll into 6 to 8 pieces. Makes 24-32 pieces