The Republicans have long been concerned about undocumented immigrants accessing public benefits. I don't mean to completely dismiss their concerns. There have been cases where undocumented immigrants have been on public benefits.
When the Republicans were in power, they mandated that enrollees in public programs document their citizenship before applying. Readers in other countries may be surprised that many Americans don't have passports, but it's true. More poor Americans don't have passports than others, and passports are a good way to prove your citizenship. People who are homeless or institutionalized, or even people who are just poor, may have severe difficulty producing passports or alternative documents like birth certificates. Working poor people who don't have documents will have to take time off work to get them replaced.
As a result, it's generally agreed that after the Deficit Reduction Act of 2005's citizenship documentation requirements took hold, a lot of citizens ended up being disenrolled from Medicaid and CHIP (the Child Health Insurance Plan), and that very few undocumented aliens were identified. Leighton Ku of the Center on Budget and Policy Priorities explains here.
He also argues that there are a number of automated data systems that can verify citizenship. For example, the Social Security Administration had a lot of data, and many states kept extensive records of birth certificates. These could have been used to verify eligibility for Medicaid and CHIP applicants. However, under the Bush Administration, the Centers for Medicare and Medicaid services issued guidance stating that applicants had to submit paper documents first. The electronic systems were to be used only as a last resort.
President Obama gave a good speech about health reform yesterday. He clearly stated his rationale for why he wanted a public plan. He addressed some misconceptions that some had about the reform plans, one of which was that they would give health insurance to undocumented immigrants. This was incorrect. However, Representative Joe Wilson of South Carolina yelled, "You lie!" at the President when he said that.
Rep. Wilson has since apologized for his intemperate outburst. However I don't believe he admitted to being mistaken about the facts. He is mistaken. The health reform bills do not allow undocumented immigrants to receive health insurance, and there are sufficient existing mechanisms to prohibit this. The Republicans have complained that they were voted down when they tried to insert an amendment requiring citizenship documentation. It's odd that they haven't heeded the lessons learned after the DRA: requiring extensive citizenship documentation hurts citizens and legal immigrants more than undocumented immigrants.
Edit: It seems to me that there undocumented immigrants may be eligible to purchase insurance, but cannot receive subsidies. They are not, as I recally, subject to the mandate. I'm not sure if there's any disagreement, but if they're willing to purchase unsubsidized insurance, they should be allowed to do so.