That's not quite how prices work. There is always a demand for labor. If the price of labor fell low enough there would be plenty of jobs. Though no one would want to work at those wages, it's even illegal to do so. The point is just that the price will never really fall to zero, even if it gets low.
>If everybody stops working, the price of services skyrockets. But the problem (soaring cost of services) and the solution (soaring reward for labor) are one and the same. This is how the free market is supposed to work. In fact, I'd go so far to say that BI presents a closer approximation to a free market than the system we have now (where the government price-controls labor).
That's not the intention of guaranteed minimum income at all, and not a good thing either. You are just cutting off one place (higher prices) to feed the other (higher wages). At least if I understand what you mean correctly, which I'm not sure. But the idea of the system is that it will balance out wealth disparities over time, but not too unfairly on people who do succeed. As well as provide a safety net. And get rid of all the bad incentives and complications created by other systems which attempt to do the same.