Both of your buttons are US Army Enlisted-men's "general service" buttons issued from 1854 through 1874. (The emblem on them was somewhat different before and after that time-period, so even without seeing their backmark we know what time-period they are from.) Also, because the army was much smaller before and after 1861-1865, not nearly as many were manufactured before or after those years. So, the statistical odds greatly favor your buttons being made during 1861-65.