First... the Waterbury Button Co. has always been located in Connecticut. That company never made any buttons for the Confederacy.
Your button's eagle is sitting on a horizontal anchor, with 3 cannonballs below the anchor, and a "rope border" encircles the stars. It is a US Navy button.
Second... the lettering in your button's Waterbury Button Co. backmark is "non-serifed" lettering. On Waterbury buttons, non-serifed lettering in the backmark means 1890-to-present.
Third... in 1941, the US Navy changed its button-emblem, making the eagle's head point toward the eagle's right side (which is the left side of the button). See the info and photos on page 103 & 104 in your copy of the button-book by Alphaeus H. Albert.
Fourth... The Waterbury Button Co. changed its name to Waterbury Companies in 1945. So, by combining all of the above facts, we know with certainty that your button was manufactured sometime between 1941 and 1945. It is (very precisely) a World War Two US Navy cuff-size button.
After-posting note: Taz42o and I were typing at the same time. As my post shows, he is correct, except that your button cannot be any later than 1945 ...when the manufacturer changed its backmark to Waterbury CO'S (the abbreviation for Companies).