I can find no documented basis for the estimates. The best estimates of undocumented immigrants is around 12 million total.
There are about 12 million foreign born who have proper papers. Most undocumented immigrants are paying taxes and are having social security deducted, as their empoyer must do. As noted in a previous post, an article in the reliable journal of the Taxation Section of the American Bar Association shows that the undocumented are paying their share.

This is somewhat of a digression inserted for information on Social Security payments- Few people seem to realize that Social Security eligibility is determined by treaty with a number of countries. For example, a Canadian geologist friend of mine and his wife both worked some 20 years (I forget the qualifying figure, but they had the necessary amount) in the U. S. as foreign nationals and qualified for full U. S. Social Security payments when they retired and returned to Canada. Similarly, Americans working in Canada will receive payments from the Canadian government when they return and retire in the States (pro-rated).
Anyone on a regular payroll, citizen or foreign national, has taxes and social security deducted by the employer, and will receive benefits pro-rated depending on length of qualifying period.
Medical insurance also is tied into most employment contracts for offshore nationals regardless of their country of origin, or where they retire.