Yes, precisely BECAUSE we would be prostrate from a nuclear attack. Critical infrastructure in the region would be destroyed, as would the means of repairing same; the survivors would head for any place better than where they were.
As bad as Mexico is, it would be better than a Southwest where major urban areas and the transportation, electrical, and water networks had been destroyed by a large-scale nuclear strike. Many survivors would thus head south (and many of these survivors would actually be illegal immigrants coming back home).
In turn, Mexico would be forced to try and stem this flow, because that many people coming south all at once would overload THEIR infrastructure.
I'll be game and concede your point is plausible. However, it still doesn't mean Mexico would not absorb the US territory it lost 150 years ago---it just means it might restrict migration from its "restored" territories, by means of "internal passports," like in the Soviet Union and Communist China.