As shown in the photo, the 3-piece button is "about 19mm or 20mm, which is a non-standard size of US Army officer's button. The Albert button-book shows two 20mm varieties, but as Nhbenz noted, the back is non-sandard, being a "self-shank" back. The extra-wide rim on it means it is post-civil-war.
The other button is an 1854-74 US Army "general service" Enlisted-men's (private, corporal, sergeant) button. Those with an iron back are very rare, but do exist. It is not known for certain whether these ironbacked ones are pre-1866, or later. I can testify that they are almost never found at civil war sites. When that happens, they might be from the postwar yankee army Occupation during the "Reconstruction" period. This one, being found at the same Massachusets dig-site as the Definitely-postwar 3-piece Officer's button, is another indication of postwar manufacturing. But we may never know with 100%-certainty about that.