A lot of this is speculation. Immigration law is not in the commonly available neocon playbook. But, the proposal to man the army with immigrant-soldiers fits neocon fancies.
Traditionally, Republicans divide into two camps regarding immigration. The nativists (aka xenophobes) fear and hate all things foreign. They believe free-trade agreements are stealing their jobs and that immigrants are a disease-ridden criminal invasion force that wants to return the southwest United States to Mexico. The Main Street Republicans (aka capitalist pigs) support free-trade because it is good for business and support immigration because it drives down the cost of labor. It is a difficult balancing act for Republicans because the nativists are their loyal foot-soldiers while Main Street is the source of multiple millions in campaign donation, not to mention all of the private bribes for earmarks.
This third leg in the immigration debate belongs to the neocons who have been mostly quiet. Neocons would seem to have more in common with Main Streeters. Unlike the nativists, they have no fear of foreigners. They are internationalists insofar as the rest of the world bows to American superiority. They approve of free-trade agreements insofar as they are colonialist tool for exploiting other countries.
I can't trace this proposal directly back to a neocon think tank, but recruiting foreigners into the army feels like a neocon idea. Neocons love history. Irving Kristol once wrote that the neocon's favorite text on foreign affairs is "Thucydides on the Peloponnesian War" (ca. 400 BCE). They are certainly aware that Rome built their great empire using legions recruited from foreign lands with the promise that, if they fought for Rome, they would gain Roman citizenship. Rome, like the United States now, could not fight their many, many wars with Roman soldiers, there were not enough Romans willing to fight. By recruiting foreigners into the ranks with the carrot of eventual citizenship, Roman emperors were able to wage endless war without burdening native-born Romans.
This is why I am convinced that the neocons have tiptoed into the immigration debate. They see foreigners as a necessary source of flesh to fuel their "long war." And, if it becomes necessary for the war party to seize the country for its own good, it will be easier if the army is not burdened by the memory of high school civics classes.