I don't see where this idea has come from?
Most of the data I can find says that the British grey and black economies are either the similar or smaller than those in France, Germany and southern Europe, which surely means it would be harder to get illegal work here than in those countries?
Its similar to the benefit arguments and Britain being some kind of migrant / asylum seeker utopia, when the weekly rate for asylum seekers in France is actually higher than the UK and it is easier for them to get permission to work.
Access to health, education and state benefits for migrants is similar across both countries but again, benefits are generally more generous in France and the health system is definitely better, at least at the minute with tue current state of the NHS.
I wonder if all this anti-immigration guff about super high numbers and people wanting to milk the system is actually a self-fulfilling prophecy.
Anti-immigration advocates use the high benefit / black market work as a siren call to the cause, creating a perception that has no basis in truth.
Through the press and word of mouth, migrants and asylum seekers hear about this apparent British migrant utopia and believe it to be true, so as word spreads, more and more try to come over here, which then further fuels the perception that they are coming to take advantage of our lax policies.