Actually illegal immigrants are helping to support the Social Security and Medicare system. Most employers require a SSN for book purposes- the SSN for the worker doens't have to be valid, but it is more of a CYA for the employer. As a result, the employer pays taxes on the income earned. Even though they pay taxes that go to Social Security, illegal immigrants are not eligible to receive it. Most of the surplus is being unclaimed since the illegal immigrants are either working with bogus SSNs or with their valid ITIN. It's estimated by SSA that 3.8 million households headed by illegal immigrants generated $6.4 billion in Social Security taxes in 2002. The Social Security Administration stashes these earnings in the 'earnings suspense file'- in the 90s, this file was worth $189 billion, twice what it was in the 80s. In the current decade, the file is generating, by year, over $6 billion in Social Security tax revenue. It is estimated that this contribution is about 10% of the surplus accrued yearly by Social Security taxes.
So there are a couple of things to take from this: Social Security is already in trouble, and from all accounts, the SSA is counting on the earnings of immigrants to help keep the SS afloat; the government, through the unclaimed SSNs, knows who employs the illegal immigrants and where the areas are that are benefitting and could crack down if necessary.
I think that the illegal immigration is a more complex issue than perhaps what is presented. While immigration is a drain on community resources such as education/health care, it is also a huge boon to SS, medicare and business.
*All stats are from the April 5th, 2005 article in NY Times.