The problem isn't illegals. It's Obamacare set expanded Medicaid up for failure.
This page displays an interactive map of the current status of state decisions on the Affordable Care Act's Medicaid expansion. Additional Medicaid expansion resources are listed (with links) below the map.
www.kff.org
Specifically, the federal government committed to covering 100% of the costs for the newly eligible expansion group for the first three years (2014-2016). This federal share was then scheduled to phase down gradually, eventually settling at 90% for 2020 and all subsequent years. This 90% federal contribution is significantly higher than the regular FMAP rates states receive for traditional Medicaid populations.
The purpose of this enhanced funding was clear: to make expanding Medicaid financially appealing for states by ensuring the federal government bore the vast majority of the cost for covering the new population.
How the Affordable Care Act's Medicaid expansion group works, which states adopted it, why the coverage gap exists, and how to check where a state stands.
govfacts.org
The Democrats did this and now they're blaming the Republicans for carrying though with it rather than eating the costs.