Background: Previous studies have shown that symptoms of depression and anxiety were highly prevalent among health sciences students. This may lead to other professional and personal difficulties and a decrease in individuals’ well-being. This study aimed to analyze levels of depression, anxiety and subjective happiness among health sciences students in Croatia. Methods: We conducted a cross-sectional study in 10 higher education institutions in Croatia during March 2023. Eligible participants were health sciences students. Participants filled out an online survey consisting of sociodemographic questions and validated scales for determining the levels of depression (9-question Patient Health Questionnaire, PHQ-9), anxiety (General Anxiety Disorder 7-item scale, GAD-7), and happiness (Subjective Happiness Scale, SHS). Results: Of 7460 invited students, 2137 students participated in the study (29% response rate). There were 41.4% of students that exhibited at least mild depressive symptoms, with 8% of students exhibiting moderately severe symptoms and 1.8% severe depressive symptoms. Mild anxiety was found in 36.8%, moderate anxiety in 23.9% and severe anxiety in 15.8% of students. The median SHS score was 19 (15.25–22). Women students had significantly higher levels of depression (p < 0.001) and anxiety (p < 0.001) than their men peers. Students in earlier study years showed higher levels of depression, anxiety and lower levels of subjective happiness compared to those in later study years. Students with lower self-assessed financial status had higher levels of depression (p < 0.001) and anxiety (p < 0.001). Students that failed an academic year had higher levels of depression (p < 0.001), but lower levels of anxiety (p = 0.005). Conclusion: In this study, we have shown that health sciences students exhibit high levels of depression and anxiety, at rates exceeding those in the general population reported in other studies. Our results may help educational institutions to put greater effort into the battle against mental health stigma, foster acceptance of mental health issues and encourage students to seek help when needed. Adequate mental health services are needed at universities to promote timely diagnosis and treatment of mental health problems.