Place eggs in a saucepan; cover with water. Bring to a boil, remove from heat, and let eggs stand in hot water for 15 minutes. Remove eggs from hot water, cool under cold running water, and peel.
Cut each egg in half lengthwise. Place yolks in a bowl; mash with a fork. Stir mayonnaise, horseradish, Dijon mustard, garlic, pickle juice, and vinegar into egg yolks until smooth; season with salt and pepper. Spoon the yolk filling into a resealable plastic bag; snip off one corner of the bag.
Place egg whites cut-side up on a serving platter; distribute minced green onion and tarragon evenly over the cavities. Pipe filling into egg white halves, sprinkle with paprika, and garnish with whole tarragon leaves. Cover deviled eggs with plastic wrap; refrigerate until ready to serve.