I suspect programs like Head Start and the school reduced/free lunch program account for a significant portion of the reported welfare use. Do you think that a poor immigrant family with working parents that uses these programs is a "net drain" on society?
Given that Head Start doesn't even have any measurable benefits, I definitely think those consuming it are a drain (if they pay less in taxes than Head Start costs).
Given that Head Start doesn't have any measurable benefits, we should conclude that Head Start is a net drain. It doesn't follow that those consuming it are. Head Start is probably an insignificant portion of the costs society invests in these people, and taxes are probably an insignificant portion of the benefit these people provide.
http://en.wikipedia.org/wiki/Head_Start_Program#Effectivenes...
http://www.acf.hhs.gov/sites/default/files/opre/executive_su...
(See also this comic if you want to point to a few blue and yellow cells in their charts. http://xkcd.com/882/ )
http://cis.org/immigrant-welfare-use-2011
Don't get me wrong - I'd love to do what you propose, though I think a BJ is a better policy than BI regardless of immigration. But I'm not aware of any western society that actually manages to do it.