It can be effectively argued that Congress has been granted the authority to require that individuals purchase health insurance under the Commerce Clause of the Constitution. (Article 1 Section 8). I’ll spare you the google search:
Section 8. The Congress shall have power to lay and collect taxes, duties, imposts and excises, to pay the debts and provide for the common defense and general welfare of the United States; but all duties, imposts and excises shall be uniform throughout the United States…
Levying a tax penalty against citizens who fail to enroll in a program which provides for the “general welfare” falls under the purview of this clause. Of course, this authority has been challenged and has so far been upheld by SCOTUS over the last several decades. Take a look at Hospital Bldg. Co v. Rex Hospital Trustees (1976). But perhaps a little more directly, the establishment of Social Security was challenged and upheld by SCOTUS in Helvering v. Davis (1937) which arguably also gives Congress the power to establish a national social insurance program.
For some scholarly research on the matter, Georgetown has a link to some interesting papers on the subject.
<link>
While President Obama is likely aware of this Article since, as you point out, he was a Constitutional Law Professor, this clause of the Constitution addresses the authority of Congress, not the President. So his legal team wouldn’t necessarily be called on to address it.