-Chilaquiles are a separate dish and in 35 (the other seven spent in the Valley and at A&M) years of living, working, and living & working in Mexico as well as traveling extensively to every Mexican state I have never seen chilaquiles served with eggs in them. On the side, absolutely and commonly. To me, even putting chicken in chilaquiles is heresy.
-FYI, true chilaquiles should not even be fried, they should be made with stale and hard corn totillas cut into length wise strips. Not the triangle shaped fried tortilla or totopo version, which has now even become acceptable in Mexico. Chilaquiles originated, just as tortillas soup did (which I never had with chicken in Mexico), by necessity to do something with left over and stale tortillas. To this day in Mexico most corn tortillas are sold wrapped in blank newsprint and or even in newspapers sometimes no bags and no refrigeration of tortillas. When I was a kid you could only buy a "kilo's" worth and done like they are they would go stale fast, thus something needed to be done with them to utilize them. My grandmother would leave out tortillas overnight to go hard and then cut up in strips and mix with oil, a brief glazing only, and sauce and serve with crumbled white cheese, onion slices, and cream. Those are chilaquiles.
-Migas are the Tex-Mex substitution/invention for incorporating Mexican chilaquiles, sans the wet sauce, with eggs in one fell swoop, or at least that is my story and I am sticking to it. I never have seen migas in Mexico, although by now I am sure they are cross polinated.
-Sorry to come across pointed on this, but it is a sore subject with me as my North Carolina raised Mexicana wife and I argue about chilaquiles at least 10 times a year. No, she does not make them either, she just likes the fried tortilla chip kind. Oh, and chilaquiles are one of my most favorite things to eat, I love them and usually eat them several times a week or month.