This is not bad, but I do have a few comments:
First of all, I don't think "restricted from free health care" is how I would put it. The word "restricted" may not be strong enough or clear enough for what you are saying. Perhaps "Illegal aliens should not be given free health care" or "Illegal aliens should not receive free health care."
Second, I wouldn't use the word "debit to" -- it just doesn't sound natural to me. I'd say "burden on."
But those are both just stylistic things. Here's my major issue here:
I don't think you should mention the school system or (maybe) the welfare system. Your thesis is about health care, not about schools and welfare. Health care is, for the most part, separate from those other two systems.
You might want to argue that providing free health care is bad because it encourages illegal aliens to stay and that puts a burden on welfare and schools, but the health care itself does not burden those systems directly.
So I'd edit this to say "Illegal aliens should not be allowed to receive free health care in the United States because that care is a tremendous burden upon the health care system."
Or maybe "... because it encourages them to stay in the country where they are a burden upon the health care, welfare, and school systems."