Efficient and secure cross-border eHealth data exchange has been recently identified by the European Commission as one of the top-three priorities for the digital transformation of health and care in the European Union. To this end, various organizational, legal, ethical, and technical challenges, related to citizens' privacy and health data security arise. This paper discusses an online survey that was conducted with the participation of European citizens, aiming to identify how they feel about exchanging their health data with healthcare professionals or eHealth service providers and to what extent they are aware of the privacy, legal, security, and technology acceptance issues (e.g. use of biometrics, mobile apps, etc.). The survey rationale, structure, and results are presented, while potential barriers and facilitators regarding cross-border health data exchange and the adoption of eHealth solutions at large are discussed.