Whisk the eggs, milk, squash puree, nutmeg and salt together until combined.
Mix in 1 cup of flour at a time until you get a batter the consistency of muffin batter: Thick and gloopy, loose enough to pour slowly, but firm enough to not immediately run through the tines of a fork.
Bring a large pot of salty water to a boil and set up another large bowl full of ice water.
Using a spätzle maker, a colander with large holes, a potato ricer with large holes, a food mill with a coarse plate, or a large-holed cheese grater, move the batter over the holes to drop little bits into the water.
They are ready when they float. Skim them off with a slotted spoon or spider skimmer and drop the spaetzle into the ice water.
When they are all done, pour off the ice water and arrange the spätzle on a sheet pan. Toss with a little oil and set aside. To serve, you can either put the dumplings in broth or saute them in fat or oil.