I don't support their using a system for which they haven't paid. I do support making assimilation and legalization of immigrants an easier and more streamlined process. People arriving illegally from other countries for whatever reason often need medical attention urgently. If we ignore them until they have passed all the needless bureaucratic hurdles, some might die. Any person dying due to an otiose system is too many, in my opinion.
Don't let them in, and there won't be that problem.
Yes, that is exactly what you are proposing. They have not paid into it, yet you advocate them using it.