You could require a magic hat and that won't solve the illegal hiring. The problem there is that nobody checks the documents. You are only required to take them, photocopy them, and file them. Then when you get a letter from the government about an error with one of the SSN numbers or something there are no consequences if you don't respond. You can just ignore it.
How would an illegal get a National ID card though? I guess the same place they get the stolen SSN's...still it would maybe help solve both these issues.
We allow illegal immigration.
Absolutely. The masters (extremes of both sides) want it, and hence the Politicians beholden to them assure it continues so as to stay in office.
I get the sense that we allow voter fraud too.
Maybe we could use some national encryption, get a dedicated employment and voting system, and go with a retinal scan then. Fed can set them up in each post office, pump some tax payer money to them for having it there, and people can interview, decide on their hire, open up an employment ticket under their business tax ID with the persons national ID (SSN could be used), the person can go get scanned at the post office - after producing a valid ID and a form from the place that wants to hire them, and then get retinal scanned. If they match, then that is stored in the system and the employer can see that the person is legit (retinal matches the national ID/SSN) and can be employed. If it doesn't match, then the person wanting to be employed better get it figured out. Obviously for hire and fire in a day work, this wouldn't help...but then, nothing would really help there anyways so who cares...
Same thing for voting. Voting location has retinal scanner in each voting booth. Person can walk into any voting booth anywhere in the country (heck, even out of country as long as the location was on the system) on national election days, scan in, vote, commit their vote, vote is electronically recorded and printout generated in booth; person is provided with their printout, other printout is recorded on paper roll in secure storage on voting machine. At same time, person is excluded from voting in that election anywhere else in the country. Since the persons retinal is tied to a current address, that vote is recorded for that address.
Sounds pretty easy to me. Should just have to go through the pain once, then it's just simple maintenance as people turn 18 and whatever (hopefully small) overhead to keep it going.