First button is a one piece riflemen coat button that is pre-Civil War era. Second button is a two piece eagle Infantry button worn by union soldiers from the Civil War.
FYI, to help with your identification requests, in the future provide the diameter of the button in millimeters. If you don’t have access to one, you can get a digital caliper from a place like Harbor Freight or Amazon for a few dollars. The size of the button will help with determining if the buttons are from a coat, kepi, vest or cuff. Also, provide the backmark information, ie the backmark of the riflemen button says Imperial Standard (I can’t read the backmark of the Eagle I button). Backmark info is extremely helpful for determining the manufacturer which leads to dating them. Hope this helps and wherever you found these buttons, go back and search for more. Congratulations.