1. Recall the trend of electron gain enthalpy

Electron gain enthalpy generally becomes more negative across a period and less negative down a group, but there are important exceptions due to very small size and electron-electron repulsions.

2. Check Statement I: "Fluorine has most negative electron gain enthalpy in its group."

Fluorine belongs to Group 17.

Expected trend might suggest fluorine should have the most negative value, but actually:

• Chlorine has more negative electron gain enthalpy than fluorine.
• Reason: fluorine is very small, so the added electron enters a compact $2p$ orbital where electron-electron repulsion is high.

Thus, $\Delta_{eg}H(\mathrm{Cl}) < \Delta_{eg}H(\mathrm{F})$ (in the sense that chlorine is more negative).

So, Statement I is false.

3. Check Statement II: "Oxygen has least negative electron gain enthalpy in its group."

Oxygen belongs to Group 16.

In this group:

• Sulfur has more negative electron gain enthalpy than oxygen.
• Oxygen, because of its very small size, has higher interelectronic repulsion when an extra electron is added.

Also, among the commonly compared elements in Group 16, oxygen shows the least negative electron gain enthalpy.

So, Statement II is true.

4. Evaluate the options

• A: Both true $\rightarrow$ Incorrect
• B: Both false $\rightarrow$ Incorrect
• C: Statement I false, Statement II true $\rightarrow$ Correct
• D: Statement I true, Statement II false $\rightarrow$ Incorrect

5. Final answer

The correct option is: $\boxed{\text{C}}$