Children of undocumented (illegal) immigrants who were, like their parents, born outside the United States have no more rights to U.S. citizenship than their parents do. Despite various legislative proposals, the U.S. immigration laws currently contain no special provisions based on the age at which a person arrived in the United States.
However, children of undocumented immigrants who were born in the United States become U.S. citizens automatically. The parent(s)' immigration status is not taken into account. This is due to the 14th Amendment of the U.S. Constitution, which reads that:
All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and the State wherein they reside.
Citizenship is the highest status available under the U.S. immigration laws. As a native-born U.S. citizen, the child in question will be granted all of the rights that every other citizen is entitled to, such as the rights to vote, assume public office, and be immune from deportation (removal).
Yes, there are many people within the United States who believe that children of illegal immigrants should not be given U.S. citizenship status. They argue that allowing such children citizenship was not the original intent of the drafters of the 14th Amendment (who didn't even address the topic of immigration, because no limits then existed on who could enter the United States in the first place). So far, however, no actual change is in the works.
Although the parents of U.S. citizens are considered their "immediate relatives," the citizen cannot use this as a basis to help the parents immigrate until the citizen is age 21. This is obviously a long time to wait.
If you are planning for the long term in this way, however, you should check in with an immigration lawyer, both now and regularly before the child turns 21. The reason is that, under current law, if you are living in the United States unlawfully after an illegal entry, you face some major hurdles to actually receiving a green card (getting lawful permanent residence) through the child. The first thing you should know is that, due to your illegal entry, you won't be able to stay in the U.S. and apply for the green card through the process known as "adjustment of status."
Why is that a big deal? Because once you leave the United States to use the alternate application procedure, called "consular processing" (where you attend your green card interview at a U.S. consulate in your home country), you become subject to a ground of inadmissibility popularly known as the "three- and ten-year time bars," whose effect is as follows:
Unless you're subject to the permanent bar, you can actually work off your time-bar period by leaving the United States either three or ten years before your green card interview (or, roughly speaking, before your child turns 21). But you'd obviously want to carefully discuss such a strategy with a lawyer first, particularly in case the laws change again.
If you have a child in the United States and either you or your spouse are within the country illegally, consult with an experienced immigration attorney for a personal analysis and to help you to explore your options.