There are two common barriers cited in discussing what prevents Americans from eating healthier: cost, and convenience. This issue takes an interesting twist in the face of immigration reform and who is allowed to work in the United States. It's possible that migrants play a positive role in the consumption patterns of Americans, and limiting their presence would make us an even larger (waist-wise) nation.
While the statement "they do the jobs Americans won't" is largely correct, it requires an addendum: Americans won't do those jobs at the wages those jobs offer. If picking cherries offered $15/hour and health insurance, college graduates would be sending in their resumes to the orchards of northern Michigan. But that's not what menial farmwork pays. And if it did, the cost of produce would rise. This is a second-level consequence of further constraining immigration we generally recognize.
Low-cost produce obviously doesn't help with the convenience issue - supersodium frozen dinners and fast food meals will always be more convenient than steaming your own vegetables. But as long as those vegetables are affordable, more people will buy them.
Would the Great Trump Wall and tightening of legal immigration actually make the county less healthy? Probably not any more than a marginal degree. It's still an interesting unintended consequence to consider.