Preheat oven to 325 degrees and spray an 8 inch square baking pan with non-stick cooking spray. In a large bowl using a hand mixer, blend the eggs, oil and water. Slowly blend in the dry brownie mix and stir until a thick batter forms.
Pour batter into prepared pan and bake for 20-22 minutes, until a toothpick inserted into the center of the pan comes out clean. Allow brownies to cool completely on a wire rack.
Use a fork to crumble the brownies, then add the Bailey's Irish Cream and peppermint extract and combine. Use your hands to form the mixture into 1 1/2 inch balls then place on a baking sheet lined with parchment and freeze for 30 minutes. Remove brownie balls from the freezer and insert a lollipop stick into the center of each one. Freeze 30 minutes more.
Melt white chocolate over a double boiler until melted. If you do not have a double boiler, place white chocolate in a microwave safe bowl and heat in 15 second increments, stirring after each time, until melted and smooth. Place candy canes in a plastic bag and crush with a rolling pin, then evenly distribute the crushed candy onto a small sheet pan or plate.
Dip brownie pops into the melted chocolate and gently tap off excess chocolate. Place the coated brownie pops onto the crushed peppermint and sprinkle additional peppermint onto each pop. Place each pop onto a sheet pan and place in freezer for 5 minutes to set the chocolate. Serve at room temperature.