On another board I was told that the US government is able to (or does) give it's citizen's start and end dates of previous employment to employers, based off your taxes. Is this true? If so, can you provide some sort of reference to back this up? If the US government does this, wouldn't it be an invasion of privacy?
No, that's patently false.
From tax history project:
Individual income tax returns — including those of public figures — are private information, protected by law from unauthorized disclosure. Indeed, the Internal Revenue Service is barred from releasing any taxpayer information whatsoever, except to authorized agencies and individuals.
Social security is built upon points, so the government does keep track of how much you worked in your life but it is all based upon what your employer reports.
The government only tracks things for three reasons: taxes, fines, or removal. At the end of the year your employer reports to the government (IRS) how much you made and how much taxes taken out, etc. They give you, the employee, a W-2 that you then report to the IRS. The IRS then matches what you report vs what was reported to them, and if all is good, then it's good to go. Otherwise you pay a hefty fine and since the government knows where you work, since the employer reports it, they'll come after you or your bank accounts. One way or another, you're going to pay them what you owe.
While they can't directly figure out start/end dates, they can determine it via when your last employer stopped reporting vs when a new one started. So they could build upon that but I see no reason they would want to do this. Unless you're receiving benefits from the government, I see no reason why they would want to track this.