Since I'm bored...
First off, to answer the basic surface question, they're illegal. The law should be enforced as written, and they should be deported immediately (Or maybe stop them from getting past our borders in the first place, but I digress). The question becomes much simpler when you actually enforce your laws. And as stated before, best not to confuse "rights" with "entitlements".
However, if we're talking about immigration and border POLICY, then immigration needs to be simplified. Not shortened in length, but merely use a straight-forward system. The main requirements that need to be expected of immigrants are...
1.) That they're moderately healthy. Essentially, we don't want immigrants bringing in any very nasty diseases, or life-threatening illnesses. This protects the general well-being of those dwelling in the United States, as well as prevents a situation where doctors will have to make a decision to treat immigrants or not with said illnesses, thus putting us into a situation where we have to force doctors to make decisions to treat said immigrants or not ( and further increase the costs of our medical infrastructure).
2.) That they have some manner of their own wealth, or someone within the United States, whether they be some sort of Non-Profit, Corporate, or Family sponsor, that's willing to support them until they get on their own feet. The oh-so-many entitlements in this country that we have are threatened with our current economic situation and debt, and we needn't have the systems in place burdened even more by people that weren't born here or haven't proven their allegiance. The reason a sponsor is needed is simply due to the inefficiency of our entitlement programs to check identification. I assure you that if you roll in the dirt a bit and put on some scruffy clothes smelling of dried urine, you can walk into your local Welfare department without any ID, claim you're homeless, and walk out with $250 of EBT for the month (or at least in California). I happen to have met people that do this, and use the EBT at crooked liquor stores to get booze for a party.
3.) That they have some ability with the English Language, or that their sponsor does, and if they're not proficient, they should be directed to where they can go to improve their capability with the English Language. They should then demonstrate competency on a final citizenship exam. We can argue all day about if English should or should not be the national language of the United States or not, but the fact of the matter is, English is the major trade language used, and we don't need immigrants to be in a lesser position than they already are, since they need to be able to do business in a mostly-English-speaking environment, and further be less susceptible to being scammed. Since this is primarily a video game site, we can take a bit of inspiration from Japan's immigration policy in this regard.
4.) That they should be educated on the essential history of our country, especially our major rights and laws, over the course of their naturalization, and should be tested on it. If immigrants become more knowledgeable about our history and rights than those that are born here and take it for granted, than so be it. Our country has an amazing history and pioneered an amazing way of living, and immigrants should be knowledgeable of this.
There could be various factors to shorten the time it takes for naturalization for exemplary candidates, but it should be around 7 years, so that enough time is given by the proper authorities to vet immigrants as necessary, and as resources permit them to.
We could probably cut the current system down to, oh, say a single hundred page bill. Or less, really, but Congress does favor its thick bills. Did I leave out anything?
Moving to the other part...
...we could improve the security on our borders at the same time. The Great Wall of Mexico (or equivalent measures) wouldn't be such a bad idea, if you also went and paired it with immigration reform. I don't believe that most people think that it's a bad thing to have secure borders, and those that do are most likely just bogged down with partisan politics.
We could do both. We just simply we need to apply enough pressure on Congress to fix the problem rather than make it another partisan battleground. But that might be wishful thinking...
Welp, boredom cured. Have fun arguing on tae interwebz.