My original question was in regards to how do those who rail against big government and government regulations reconcile the fact that the status of "legal" or "illegal" is defined by the government,
Then the simple answer is that this is not just a function of "big" government. It's a function of even the smallest government: to defend the populace against external threats. And since global or universal governments are wildly impractical, that will not change any time soon. But I'm not an anarchist, so I really can't tell you what those who rail against small government think.
and the government regulates whether we can employ or house illegals.
Yes, which is an unfortunate consequence of the particular wealth redistribution policies preferred in the US and other countries. Yet as I pointed out, these policies favor immigrants
by redistributing wealth away from natives and into the market. So if you want to get rid of them, you'd probably end up with much less economic incentive to employ or house illegals anyways and the point would be near moot. But, hey, I'm all for it. It's just a bigger job than you think.
In respect to big government, regulations, and free markets, it doesn't matter what illegals contribute to the economy; just that they can, and no one should be able to make them not.
But they can also take away from the economy, which is the reason they are regulated. And since, for completely undocumented immigrants, it takes at least a generation to determine whether they are likely to be a net gain or a net loss, they should be heavily regulated.
Nor, of course, should anyone be forced to support them if they don't.
Regardless of whether you are forced to support them, you are forced to compete
with them. And this has almost the exact same effect, potentially even a total race to the bottom in terms of living standards in the case of completely unrestrained immigration from actively malicious or totally failed foreign states. But like I said, feel free to change this by starting with the biggest government of all, the FED.
As for what they help to save, what do you mean by "save?"
I mean what resources are saved? That's what "economy" means -- savings.
You've been brainwashed by FED propaganda into believing that "economy" is synonymous with "consumption". That's the official line. It's built into the GDP. That's how the talking heads can say with a straight face that spending, and debt, and warfare and immigration are all "good for the economy." Well, they aren't. That's nonsense and if you're going to repeat it, you should back it up.