Fill a medium saucepan halfway with water and bring to a boil. Carefully add the eggs and boil for at least 10 minutes. Remove the eggs and place in ice water to cool.
Gently crack and peel off the shell. Cut the eggs in half lengthwise.
Spoon the yolks into a small mixing bowl and mash with a fork until smooth. Mix in the mayonnaise, mustard, relish and minced garlic. Season with the Mesquite seasoning. Sample the mixture and adjust any ingredients to taste, if desired.
Spoon or pipe the yolk mixture into the eggs whites. Cover and refrigerate for about 30 minutes or until chilled.