First button:
Having an iron back and weakly defined eagle-standing-on-anchor emblem causes some people to think such buttons are Confederate-made. However, no Historical Document evidence has surfaced that the Confederates every manufactured any copies of that US Navy emblem on 2-piece or 3-piece buttons. Your button is not shown in any of the books on civil war Military buttons. (The book by Warren Tice does show a 1-piece "solid cast white-metal" version.) Therefore, we button-collectors believe the type you found are crude imitations of the US Navy button, manufactured for use on Civilian clothing, such as a blazer jacket or sportcoat. They start showing up in the late-1800s ...and are still being manufactured today. (See the Jo-Ann Fabrics & Sewing-Supplies website and stores.)
2nd button:
It is definitely an 1854-to-1874 US Army Enlisted-man's button. (The term "Enlisted-man" means the ranks of private, corporal, and sargeant ...buttons for Officers were fancier.) Diggers call the version you found simply as an "eagle button." It is the most common civil war Military-issue button.