There is no junk or processed food in our house, no breakfast cereal either. I try to have no junk at all in lunch boxes but will have a home cooked baked item where I can control what's in it. But on birthdays, parties, special occasions, Nanny's house etc I am happy for them to have treats and enjoy them. We have fish and chips on Sunday nights as a rule.
I hope we are teaching that if every day food is healthy then you can have treats when appropriate and that's ok.
I still get the eye rolls and comments from extended family and friends but I am learning to ignore!!