That is a lie that is being perpetrated by those who want the illegals in our country.
For example, we have OSHA and all kinds of rules that make all jobs relatively clean and safe. The reason they want illegals to do the work is because they are behaving illegally and if the illegals complain they simply round them up in the night and ship them back across the border.
Many of the jobs they do are in meat packing industry. You can get people from every race and ethnicity wanting to work there as long as they are following the laws and standards of OSHA, so it isn't that they are doing jobs that others don't want to do, it is that the business owners don't want to follow the laws and so they need people who can't complain.