If they are not 'subject to the jurisdiction thereof', how can they be illegal aliens? Simple. They are "subject to the jurisdiction" of a foreign power, not the US, hence they're not US citizens, but citizens of that country whose jurisdiction they are subject to.
If they are in the US and not on the short list of exceptions, diplomats, Indians, Hostile Occupiers, etc. then they ARE subject to the jurisdiction of the United States. They can be arrested, jailed, tried, convicted, deported, given in-state tuition, get speeding tickets and not pay them, get evicted, collect welfare, sue people, yada yada yada. They do not have diplomatic immunity. Here, read
this. Second paragraph has some meat.