This is nonsensical scaremongering. The UK's refusal to join Schengen keeps out a handful of illegal migrants, most of whom would be caught and deported anyway. If the far greater number of French[1] and Eastern European migrants who live and work here courtesy of the EU's "free movement of labour" clauses haven't "overwhelmed" anybody,[2] then a few extra illegals are hardly going to tip the balance.
[1] - Oddly, nobody ever complains about the huge numbers of French migrants to the UK. It's almost as though there's an element of racial discrimination somewhere...
[2] - Boston, Lincs. is often cited as being overwhelmed by Eastern migrants. But even the BNP can only find an uptick in driving offences to complain about: http://www.bnp.org.uk/news/regional/stop-press-boston-crime-...