That's the common refrain among business owners and farmers, ranging from Trump's Mar-a-Lago operation to a medium size dairy operation. Liberals like me tend to buy the statement, because we're usually in favor of immigration, so the statement operates as justification.
When you think about it, though, it's unusual for liberals to trust Trump or other business owners. :-)
Why should we think the statement is true, why are immigrants willing to work off-hours and the worst jobs? I think one reason is found in reference group theory, which is the sociologist's jargon for saying "everything is relative". Immigrants compare their work and working conditions in the U.S. with what they faced in their home country and find it not so bad. The American-born compare the same jobs with other jobs, and know they're the worst.
There's also the relativity of compensation: immigrants will find that the salary and possibly fringe benefits far exceed that of their origin country. I suspect there's a human tendency to focus on the rewards and not the cost of living. The American-born will find the salary toward the bottom of the scale.
There's also the standard of living: an immigrant can see crowded living conditions in a less-desirable neighborhood as still being a step up from home. The American-born would likely find the conditions among which some immigrants live as not desirable.
And finally there's the time frame: the American-born looks at the less desirable job as a dead-ender. The immigrant can view it as a step up for the future, whether it's moving from dishwasher to prep work to sous-chef or simply saving money to buy goods to take back home (see Sam Quinones "Dreamland").
Among those who want to reduce immigration the standard reply to the statement is: "raise your pay."
I think that's wrong, pay being only one of the factors which makes a bad job acceptable to an immigrant. My advice to those who would reduce immigration is this: look to the military.
The military is a case where they offer bad jobs (I'm talking basic training, which is likely worse than any normal "bad job") and attract people to them. An E-1 gets about $17,000 a year, before taxes. How do they attract people? Basically it's the promotion and the fringe benefits, the retirement and education benefits. So immigration restrictionists should come up with a program where the government provides good benefits and the possibility of advancement to the crap jobs. Tell the high school drop out, spend x months doing this job and you'll earn tuition for college, have health insurance, etc. etc. Is that proposal naive? Perhaps, but I'd like to see it tried.