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“Apparently, socialist Mexico does not have a government-provided healthcare system, so the US taxpayer gets to supply one to their illegal aliens”

Actually Mexico does have a government provided healthcare system...but like everything in Mexico, it isn’t worth squat.
And if their citizens can get free care here, why not!

CBC News - Health - How Mexico’s health system works

Here’s roughly how the Mexican private health system is structured:

* About three million wealthy and middle class Mexicans (and foreigners working in Mexico) pay private insurers to gain access to high-quality, state of the art medical services. Many Mexicans who are insured through the public insurance system also pay out of pocket for private care to get better service.
* Mexico’s private sector is booming. New clinics and specialized hospitals are growing rapidly in Mexico City, Guadalajara and especially Monterrey. In fact, Monterrey is becoming a big centre for medical tourism, particularly for Americans trying to escape their own expensive health-care system. They’re flocking to the city’s growing number of gleaming new hospitals for everything from obesity surgery and angioplasty to hip replacements. A hip replacement in the U.S. can cost from $43,000 to $63,000 but in Mexico it’s a bargain at around $12,000.

There is also a public system:

* About 50 million salaried Mexicans pay into an insurance scheme, along with their employers and the government, through the Institute of Social Security. The employee pays a progressive amount, according to his or her wage. The institute runs its own primary care units and state hospitals for insured workers, although the quality varies considerably. Critics say some establishments are not properly managed and lack adequate equipment.
* About 17 million state employees have a separate, parallel scheme that they pay into through the Institute of Security and Social Services for State Workers. They also have their own clinics and hospitals.
* The army and navy also have their own separate insurance programs, funded partly through social services, as do employees at Mexico’s state-owned petroleum company, Pemex.

Universal health care

Roughly 40 million uninsured Mexicans — the country’s poorest — all began moving toward universal access to full health coverage in 2003 through a program called Seguro Popular (Popular Health Insurance). It’s the most ambitious health insurance program in Mexico to be launched since the beginning of social security in the country in 1943, and it’s meant to reduce the inequality of health services and stop poor families from being financially wiped out by an illness. The federal government has set a target date of 2010 to provide the coverage to all uninsured Mexicans although it’s uncertain whether they’re on track.

Families pay a premium to join, based on their income, and have to make preventive health-care visits at clinics. About 20 per cent of the poorest families pay nothing.[snip]
http://www.cbc.ca/health/story/2009/05/04/f-health-mexico-health-system.html