First ionization enthalpy generally increases across a period. However, due to half-filled stability of nitrogen, oxygen has lower ionization enthalpy than nitrogen. Hence, the correct increasing order is C < O < N < F, making Statement I true.
Electron gain enthalpy depends on atomic size and inter-electronic repulsion. In group 16, oxygen has lower magnitude of electron gain enthalpy due to its small size and high repulsion, while sulphur has the highest magnitude. Thus the correct order is S > Se > Te > Po > O, making Statement II true.
Therefore, both Statement I and Statement II are correct.