For the eggs: Place the eggs in the bottom of a large stockpot in a single layer and fill the pot with water to cover the eggs by 2 inches. Add the salt. Place the eggs over high heat and bring to a boil, then turn off the heat, cover the pot and let sit for 13 to 15 minutes. Prepare an ice bath. Remove the eggs from the pot and place into the ice bath to cool.
For the filling: Peel the eggs, cut them in half and place half of the yolks in a food processor. Add the basil, chives and 1/2 cup of the mayonnaise to the processor and puree until smooth. Season with salt and pepper. Transfer the filling to a resealable plastic bag.
Clean out the bowl of the food processor. Place the remaining egg yolks into the clean processor, add the roasted red peppers and remaining 1/2 cup mayonnaise and puree until smooth. Season with salt and pepper and transfer to a second resealable bag.
Add the red food coloring and 1/2 teaspoon of the vinegar to 2 cups of room temperature water in a bowl and mix well. Add half of the egg whites (make sure you remove the membranes from the egg halves so they don't affect the dyeing process) and let soak until the desired color is achieved, at least 15 minutes. Repeat the same process with the green food coloring and remaining vinegar. It will take longer to get a deep green color.
Remove the eggs from the dye and pat dry. Arrange on a serving tray and pipe the green filling into the red egg halves and the red filling into the green egg halves. Garnish the green filling with chopped chives. Garnish the red filling with finely chopped red pepper.