Can a health insurance refuse us as a client?

In Switzerland, you cannot be rejected, all insurers are obliged to accept you, without any health questionnaire and no matter the current state of your health. 

The mandatory health care insurance is, as its name suggests, mandatory. However, some exceptions do exist. But since we do not have a choice, it makes sense for the subscription to a basic insurance not to be conditioned to the acceptation of the insurer. 

However, every supplementary insurance and, more broadly, every optional insurance admission is subject a stringent and painful questionnaire. 

Do not worry, with the LAMal, you will be able to go wherever you want without risking paying more than another person or not being covered for the same events.