(1) Background: Changes in daily life and academic training has led to uncertainty in the higher education student population during COVID-19. The goal of the study was to examine the impacts of the pandemic on Hungarian students. (2) Methods: A cross-sectional study was conducted by using self-report questionnaires collected in Google Forms. Eight-hundred-and-twenty-seven students (25.29 ± 8.09) took part anonymously. The respondents rate their overall physical and mental health on a 5-point Likert scale and validated scales were used: Well Being Index (WHO-5); Mindfulness Attention and Awareness Scale (MAAS); and Perceived Stress Scale (PSS-14). Statistical analyses were performed with IBM SPSS-14.24, results were considered at a significance level p ≤ 0.05. (3) Results: Positive correlation was found between MAAS and WHO-5 (r = 0.363, p < 0.001) negative correlation between MAAS and PSS-14 (r = −0.448, p < 0.001), and negative correlation between WHO-5 and PSS-14 (r = −0.671, p < 0.001). Females had higher PSS-14 mean score (32.51 ± 10.16) than males (27.71 ± 10.19; p < 0.001; Z = −5703), males (60.92 ± 12.10) had higher MAAS level than females (57.31 ± 12.51; p < 0.001; Z = −3589). No difference was found in gender regarding WHO-5 mean scores. Athletes (7.03 ± 3.27) differ significantly from non-athletes (6.00 ± 3.04) in WHO-5 (p < 0.001; Z = −4.349) and MAAS level (p = 0.012; Z = −2.498), but showed no difference in PSS-14 (p = 0.101; Z = −1.641). Students rated mental (3.01 ± 0.99) worse than physical health (3.49 ± 0.98; p < 0.001, r = 0.426) and the narrowing of social relationships worse (3.83 ± 1.26) than physical (p < 0.001, r = −0.212) and mental health (p < 0.001, r = −0.408). Females had worse mental health (2.96 ± 9.94) than males (3.20 ± 0.99; p = 0.003; Z = −2.924) and rated the narrowing of social relationships worse (3.90 ± 1.23) than males (3.59 ± 1.35; p = 0.006; Z = −2.730). (4) Conclusions: The pandemic has negatively impacted students, and it may have long-term consequences on their mental and physical health and education.