This is my concern as well. Im glad he's making the attempt either way.
Ill be curious to see more details why he believes its constitutional as the 14th amendment is not specific. I have a feeling his argument is somewhere along the lines of the 14th amendment protects citizens only, and thus, cannot be used as constitutional shelter for those not here legally.
It would get tricky from there, as there is some prior precedence. In Landon v. Plasencia (which was about due process of a resident alient), it was ruled that a resident alien was entitled to constitutional protection once that person develops permenent residence ties to the place they reside (live). So, if an anchor baby is literally born in short order after crossing illegally - before said ties could be made, he could have an argument.
Link
Excerpt
Congress has exclusive authority to establish immigration laws and procedures. 1 This authority is limited, however, by the constitutional rights of resident aliens. 2 An alien attempting to enter the United States for the first time has no constitutional rights and thus may be excluded 3 on any ground and by any procedure the legislature deems appropriate. 4 Once an alien has been admitted into the United States, 5 however, his or her status under the Constitution changes significantly. 6 When expelling a permanent resident alien, 7 the government must afford the alien many constitutional protections, including procedural due process. 8 Deportation proceedings for resident aliens have been developed by Congress in conformance with these constitutional protections. 9 In contrast, exclusion procedures for aliens attempting initial entry into the United States have had no such constitutional restraints. 10
An anchor baby, by definition would seem to fall perfectly under the first bold. So just how unconstitutional is it?