Step 1: Analyze Statement I.
Across a period, first ionization enthalpy generally increases due to increasing nuclear charge.
However, there is an exception between oxygen and nitrogen.
Nitrogen has a half-filled \(2p^3\) configuration, which is more stable than oxygen’s \(2p^4\) configuration.
Therefore,
\[
\mathrm{C<N>O<F}
\]
is the correct trend, not \( \mathrm{C<O<N<F} \).
Hence, Statement I is false.
Step 2: Analyze Statement II.
Electron gain enthalpy generally becomes less negative down a group due to increasing atomic size.
However, oxygen has a much lower (less negative) electron gain enthalpy compared to sulfur because of strong inter-electronic repulsion in the compact \(2p\) orbitals.
Thus, the correct order of magnitude is:
\[
\mathrm{S>Se>Te>Po>O}.
\]
Hence, Statement II is true.
Step 3: Final conclusion.
Statement I is false but Statement II is true.
Final Answer:
\[
\boxed{\text{Statement I is false but Statement II is true}}
\]