This is a complicated subject, and there has been a lot of research done on it. The question of does a min wage increase unemployment numbers is really incomplete without asking what the increase is? Is $15 too much? in rural Mississippi probably so, in Berkley Ca maybe not.
The example cited in the article is not a good one becasue a university or any type of agency that operates on a fixed budget has to react to a change like this immediately, so they probably have no other choice but to lay off workers, whereas a private sector company can comply if it chooses even if it requires raising prices. If your business requires 5 workers, it may not be feasible to operate with 4 workers so compliance is not optional.
The argument that min wage has a positive effect on the economy stems from the idea that putting more money into the pockets of low wage workers results in their spending the money and thus stimulating the economy creating more demand for products and services. Here is a statement from congressional report to the president:
"In recent years there have been important developments in the academic literature on the effect of increases in the minimum wage on employment, with the weight of evidence now showing that increases in the minimum wage have had little or no negative effect on the employment of minimum-wage workers, even during times of weakness in the labor market. Research suggests that a minimum-wage increase could have a small stimulative effect on the economy as low-wage workers spend their additional earnings, raising demand and job growth, and providing some help on the jobs front."
Another argument against large min wage increases is that it will drive automation which will replace jobs with machines and software. There is some merit to this argument but IMO this is going to be done anyway, in fact over the last decade more than 5 million jobs have been displaced by automation.
If you look beyond just the immediate effect of a few people getting laid off, what you see is a bigger picture that includes some growth but also reduction in tax payer burden. If people are being paid poverty wages, which is what many retailers and restaurants pay, then those people are eligible for public assistance for housing, SNAP etc. Walmart has been used as an example but many retailers fall into the same situation. Some studies showed as much as 40% of Walmart workers were collecting government benefits. Those benefits have to be paid by tax dollars which creates higher tax burden or more borrowing by the federal government.
So the problem isn't quite as black and white as it may seem. The Federal min wage should be based on the lowest cost of living areas, then if other higher cost areas want to establish a higher min wage then state and local governments can do so. IMO congress should just look at the current min wage and have an automatic escalation provision written into the law that would allow it to be raised by the percent of inflation. That would in essence solve the problem once and for all and not require continued debate and congressional action.