We don't have much of a problem with illegal immigrants in Canada, because you can't get ANYTHING in this country without a valid Social Insurance Card, and every governmentt agency requires your card and will e-verify your number.
Refugee claimants receive access to health care upon entry to Canada, but then there are no special health care services for our veterans. Everyone living in Canada has access to the same level of care - regardless of how long they've lived here. That includes health care. Basically, we greet you at the immigration office with "Hi, welcome to Canada. Here is your health card card.
One of my American friends from New York State, who studied in Toronto in the 1990's said American students living close to the border, kept their OHIP cards and frequently returned to Canada for "free health care".
Employers caught hiring undocumented workers will be fined $10,000 per employee. Canadians won't work under the table, because they can't get unemployment insurance, or government retraining assistance unless they've contributed premiums to the programs for 9 of the last 12 months.