In a large mixing bowl, stir together the ham, eggs. onions, and celery until just combined.
In a separate, small bowl whisk together the mayo, relish, mustards, and pepper until the mixture's smooth & evenly combined. Add this mixture to the bowl of ham and veggies. Stir just until evenly combined and everything's coated.
You can serve immediately, but I recommend chilling 1-2 hours first. Season with salt, to taste, just before serving. Remember ham is naturally salty, so you may not need any, and probably not much.
Serve sandwich style, with buttery crackers, or in a healthy lettuce wrap.