Gretchen-
16 mo is still plenty young, so if you're ready to commit to changing his eating habits, it can certainly be done. Remember that you set the example...if DS sees you eating junk food, he's going to want it too. I'm not saying you have to change YOUR diet, just be mindful that he's always watching. Yes, DH and I eat junk, but not in front of Tyler. Sneaky, yes, but it works for us. Treats are fine, but they should be just that...treats. Once you've changed his habits, there's nothing wrong with an occasional treat. All things in moderation!
My best advice to you is to always offer 2 choices, and make both of them healthy. Make one or both of them something he likes. If he insists on something unhealthy, don't give it to him. You're likely to get some tantrums, skipped meals, etc, but you can only change his habits if you hold your ground. At that age, you control WHAT he eats. He may well decide he'd rather not eat, but it will probably only take a few skipped meals for him to catch on. Remember that healthy kids don't starve themselves...if he decides to make it a battle of wills and go on a hunger strike, shrug and move on. He will eat eventually.
Tyler goes through non-veggie phases, but I don't make a big deal out of it or insist he eat something he doesn't want. I do sneak veggies into him at every opportunity. If you can give me an example of some of the things he eats, I'd be more than happy to help you find ways to get veggies in there!