In a perfect competitive market, there would be a single equilibrium price for low-skilled labor across the entire world. That's because the theoretical perfect market is based on the non-existence of things such as national borders, relocations costs, and perfect information. Basically, worker A would know, at all times, which companies have available positions, exactly how much value he would add to the company, and he would be able to move instantly to the highest-paying job for which he qualifies. Similarly, company 1, would know every person who is willing and qualified to fulfill an employment need (without the need for interview) and would not incur any expense in locating and hiring new employees to replace those who start to shirk their work or to meet growing demand. In such a perfect world, a minimum wage set above the equilibrium price causes unemployment.
We don't live in a perfect world. Different population centers have different supply and demand for low-skilled jobs, and there are transaction costs when workers want to relocate to available jobs, or when companies want to relocate to jobs to locations with higher demand for the products. In such an imperfect market, different population centers have different prices for low-skilled labor (different wages). Further, disparity in bargaining power between employers and employees can result in below-market wages (if, e.g., an employee is desperate for a job and can't move because their spouse works in the area will accept a low-ball offer) or above-market wages (if, e.g., an employee in a competitive labor market quits without notice, an employer might offer a higher wage to fill and urgent vacancy without a long interview and hiring process).
Let's be honest, more often than not, employers have stronger bargaining power, so the market-wage for low-skilled labor tends to be lower than the equilibrium-wage in a perfect economy. In such situations, minimum wages correct the market and will not result in higher unemployment.
However, remember, that the imperfections also result in differing supply and demand curves for different geographical areas. That means Seattle might have a different market wage than Olympia, Phoenix, Detroit, and Backwoods-Village. In this more realistic situation, raising the minimum wage might not cause any unemployment in Seattle, but it might cause people to lose jobs in Phoenix and backwoods-village. That's why minimum wages aren't great federal policy. It's better to let states and cities set them, as they are in a better position to analyze the effects and determine whether a minimum wage was set too high.
