Add eggs to a saucepan and cover with cold water. Bring to a boil over medium-high heat, then turn off the heat and cover with a lid. Let the eggs sit undisturbed for 10 minutes. Remove the eggs from the water and rinse under cold water to cool down. When cool enough to handle, peel and slice.
Toss together the tuna, red onions, cherry tomatoes, celery, cucumber, mayonnaise, balsamic vinegar, basil and some pepper. Mix to combine. Transfer to a serving platter, arrange the sliced eggs on top, and serve.