From your link:
Let’s begin by assuming that, despite what he says, Underwood isn't aiming for Universal Employment (an unemployment rate of 0% is near impossible), but is aiming for full employment. Full employment is often seen as an “ideal” unemployment rate. It reflects only some frictional and voluntary unemployment, where workers are temporarily searching for new jobs or don’t want one for some reason. In its most recent estimate, the Federal Reserve has placed full employment in the United States at somewhere between 5.2% and 5.5%.
This paragraph has several problems. Let's start from the end. Current unemployment is measured at 3.9%. Therefore, by this definition, the United States currently has more than full employment. The more common term for this is the natural unemployment rate. Example paper (PDF). Or optimal unemployment (PDF).
The current Federal Reserve estimate is between 4% and 5%. That article is out of date.
In addition, "don’t want one for some reason" is not included in the unemployment statistic. It only counts people who are actively looking for work. Someone who does not want one for some reason is not a member of the labor force.
The problem if there was actually 0% unemployment is that then there would be no one to hire. Companies would have to hire their employees from other companies, presumably paying more to make them switch. But then those companies would have to replace the departing workers. This situation would tend to lead to inflation, as companies offer more and more money to attract workers and increase their prices to compensate.
Part of the issue here is that the unemployment rate is the wrong metric. A better metric is the number of job seekers to job openings. If those numbers are equal, then regardless of the unemployment rate, the economy is in balance. If there are more seekers than openings, then there is involuntary unemployment. If there are more openings than seekers, then expect wage increases and eventually inflation.
Another issue is how many people are currently out of the labor force. If that number is large, then that can allow for more job openings to those officially seeking. Instead of companies hiring from those looking, they can recruit people who dropped out of the labor force. This does not create the same inflationary displacement effects as hiring from other companies.
How this translates to an unemployment rate is unclear. The traditional definition of full employment was 6%. But it's been dropping. One reason for this may be that it is much easier to match job opportunities to seekers now. Traditionally seekers had to scan a large number of paper records looking for compatible jobs. Now that information is computerized. People can search for jobs by location, title, and/or other keywords. And employers can publish jobs much more easily as well. As a result, the frictional costs of unemployment may have been reduced.
This also may make it easier to switch jobs. So rather than wait to be laid off, workers may be preemptively leaving. They never even show up in the unemployment rate now, because they are never actually without a job. Or in previous times, they may have needed to quit to have enough time to engage in a real job search. Whereas now they can spend only a few minutes on the search. They just need to find time to interview.
The applications processing this may continue to improve. Perhaps the traditional unemployment rate may drop to 2% or 1%.
Another possibility is that employers have become more flexible. Instead of having to hire local to their headquarters, perhaps it is easier for employers to maintain multiple offices. Teleconferencing equipment makes it easier to be in multiple cities. So when there are few potential hires in one city, they may be able to shift operations to another city where there are more people available. And that may be easier than the more traditional possibility of moving for work.
This could be true of some employers more than others. For example, telemarketing can be done from pretty much anywhere. So those employers may constantly shift to wherever potential employees are most desperate for work.