You have the perfect ingredients for an excellent meal! I would do it this way, but this is variable to one's own taste. I would dice an onion and separate into two halves. In two fry pans fry diced onion in a tablespoon of bacon fat till just browning, then add a tablespoon of flour to make a roux, fry for a couple of minutes till lightly brown. Add one can of enchilada sauce to each pan and cook till lightly boiling and thickened.
Now if you prefer crunchy chiaquiles, take pans off heat and add fried tortilla strips, toss quickly to coat and top with jack cheese, or melting Mexican cheese if you have that. If you like softened chilaquiles(as I do), add tortilla strips to thickened sauces over low heat and stir to coat, then cover pans for a few minutes and then stir again and add half of cheese and stir into chilaquiles, then top with rest of cheese and cover for a minute to melt. You now have a pan of green and a pan of red chilaquiles.
On each serving plate, place a portion of each chilaquiles separated by a row of refritos(if available and desired) and top each portion of chilaquiles with an over-easy egg. Sprinkle crumbled Cotija cheese and cilantro over each egg. Serve with salsa or hot sauce to adjust heat as desired.
Red and green chilaquiles are generally considered to be distinct dishes. They also not covered in enchilada sauce, but in some sort of salsa. If you want to take the easy route and not make your own, you would be best off buying some mild jarred salsa and tossing your softened day-old tortillas in that, then topping with a crumbly cheese.