The Declaration of Independence states that "All men are created equal", not that "All Americans are created equal". Many people support immigration in theory, but complain about illegal immigrants getting access to privileges like healthcare and education. Since they aren't subject to taxes, this seems to make sense. Why should taxpayers subsidize public services for non-taxpayers?
If you take this view, as I do, then you should ask yourself if your fighting the right battle. The US is based on the revolutionary idea that any person can come to this country, and by dint of his/her effort and talents make themselves a better life without interference OR assistance by government. The idea that we should close our borders and keep our prosperity to ourselves is extremely reactionary, and goes directly against the principles that have made us the greatest country on earth. If you are against subsidization of healthcare/education for non-taxpayers, then argue directly against the provision of these public services, not against the immigrants who have had no choice in the matter.
Any true American understands that we are entitled to Life, Liberty and the Pursuit of Happiness by virtue of being Human, not on the basis of our ethnicity, religion OR which arbitrary border we happen to be born within. For this reason I support completely open borders.