I am not THE libertarian to fully hold this argument and as others have mentioned there are libertarian arguments for universal healthcare, but I will present the best case I can from those I've heard be against it.
The primary case is the idea of negative rights vs positive rights. Where the idea that the state should protect you from others wanting limit your rights vs providing you the ability to do something.
So using the state to punish someone for who is trying to stop you from providing healthcare service is justified use of violence as it protects your negative rights and define and preserves you and the violators boundary. Whereas using state violence to force you to provide healthcare someone you don't want to would not as it violated your negative right.
This is primary argument against any positive right, is that since it requires a service to be fulfilled the state would be use violence (the basis of state power) to enforce it. Making it tantamount to slavery.
Now the reality of it though is that most libertarians do support this slavery at least in service of giving the state the monopoly on violence (police, military, etc) in order to protect their defined negative rights. And because of our current material abundance we are able to have a fractionalized slavery extracting wealth from people to small enough degree that most people don't find as aborrent full servitude of an individual.