No. Sergeant Avery Johnson appears in the **single-player campaigns** of Halo: Combat Evolved, Halo 2, Halo 3, and Halo 3: ODST. You encounter him purely through story progression.
Xbox Live Gold / Game Pass Core is only required for **online multiplayer matchmaking** (Matchmaking, Firefight matchmaking, Co-op matchmaking). Local split-screen co-op and the entire campaign work offline with a Silver (free) account.
If you're referring to **Halo 3 Multiplayer armor (Sergeant Johnson armor permutation)**: That was unlocked via **Vidmaster Achievements** (e.g., "Annual" – complete any Halo 3 campaign level on Legendary solo) or purchased as DLC (no longer available). Did not require Gold for the achievement unlock itself, but Gold was needed to download the DLC originally.