At the same time, an UBI will make it so people still don't care to perform certain below-current-minimum-wage jobs. E.g. if I were on UBI, and I effectively earned $15/hour through it, I still wouldn't bother looking at jobs unless they paid, say, $5/hour or more.
I think that's one of the most fascinating questions about UBI though. Hypothetically, if you had UBI and were content at that level of consumption, what would you do? If I absolutely loved animals, I can certainly see working in a pet store for $2/hr or a pittance.
To me, minimum wage says "if you work, then you should at least able to support yourself on your income$." UBI+ says "if you work, you must find greater fulfillment than mere sustenance in your work."
Personally, I can't see minimum wage applying to anyone receiving UBI. Although it might make sense to have a "less than minimum wage" tax that helps fund the UBI directly.
$ Abstracting away actual amounts and whether or not they're enough
+ Market adjustments aside