The arguments are well-founded (although I disagree with them), and speak out against a far more specific issue than Immigration.
If the arguments against allowing immigrants to join the army were:
  • Foreign recruits have less loyalty to the USA so may not fight as hard
  • There may be language, dietry and culture differences that may lead to problems
  • The troops may be deployed in their home country leading to conflicts of loyalties


Then you could argue that these arguments are well-founded (although I would disagree with them too). However the arguments I found against the idea on this thread concerned the chances of these immigrants being terrorists. This is far from "well-founded", and in my view, betrays a prejudice against immigrants. I am not sure that Americans look as favourably on current immigration as you suggest, but American attitudes are far better than British ones.