~snip~
Illegal immigration results from weak laws and political parties who support it for short-term electoral gain. The rules in USA and EU are very similar and it is very difficult to deport illegal aliens unless they commit crimes. We have a similar situation here in India, where the supreme court frequently blocks any attempt from the government to deport illegal immigrants (mostly from Bangladesh). On the other hand, countries such as Australia, Saudi Arabia and Japan doesn't have this issue, because the laws are enforced strictly. And this is also one of the reasons why these countries are having a lower crime rate compared to the USA.
There's also a huge incentive for businesses to keep low paying workers in the country. It's probably not for the kind of their hearts that they keep illegal immigrants in the country.
Every restaurant/pub kitchen in the US is basically manned by someone from another country. Maybe some of those are paid a bit less than what they should be paid.