Are expats eligible for the public system? What is the quality of the public system? How are the costs for the private system? (compared, for example, to those in the US)
Expats living in Chile are eligable for public and private healthcare.
Public healthcare in Chile is provided by the state-funded FONASA scheme.However, although the standard of public healthcare is fine in Santiago, the standards do vary regionally. For this reason, most expats prefer to opt for private healthcare.
You will need to look at various international health insurance providers and compare rates. Certain things like international evacuation from Chile can be pricey so just be aware to make sure these are included in your price.
For more information check out the Healthcare in Chile page