The animal shown at 9-o'clock appears to be a rabbit... long upright ears and a cottontail, compared to the short ears and long tail on the dogs. Your button shows a game-animal (or perhaps fox) hunting scene, so it is a civilian-usage type called a Sportsman's or Sporting button. According to the two button-dating charts attached below, your button dates from the 2nd half of the 1700s into the very early 1800s.
Are you sure the metal is yellow (meaning brass or bronze)? Its construction looks more like the cast "white metal" buttons shown in the dating-charts.
Might have been helpful for you to tell us what the writing on its front says.