In other words, can you really have BI without universal health care?
Yes, of course. Universal health care coverage (read: insurance) can't be used to buy groceries or pay rent. Basic income will never be enough to pay for somebody's cancer treatments.
Neither one is a substitute for the other; both are extremely beneficial.
Currently under the PPACA you cannot be denied insurance for prexisting conditions, and it is reasonable to expect pegging UBI to just poverty + cost of bronze single person plan in your area rather than implementing universal healthcare.
Not trying to argue it isn't cheaper to cut out the insane middle men in the healthcare industry, or that the current insurance system works at all (deductibles cost society incredible amounts of money and time because people won't go to the doctor until they are dying because of the unsubsidized costs) but if your goal is to have everyone insured, fed, and sheltered you can do that with just the cash payments while just enduring a disastrously inefficient and harmful healthcare system besides it.
It depends on your definition of "basic".
I normally take "basic income" to mean "covers typical recurring monthly expenses that almost everyone has".
If that's the definition, then I believe you can.
With health care taking care of medical expense emergencies, basic income is left to take care of job-related emergencies. Your example of "my car broke down" would seem to fit that category. So what then? Joe loses his job but basic income will make sure he never goes homeless and always has enough food to eat. Instead of having a major crisis he can take a break from working and maybe upgrade his skills.