I don't suppose I'll be much help, but it's a bit quiet at the moment so I'll try! I would say that if you think that squash might be a problem it's much easier not to have it in the house, so it really is juice or nothing.
Two things which I found helped when dd was that age (and very faddy) were home-cooked versions of junk.
Instead of chips, cut potatoes (ideally Charlotte) into small cubes (about 1 cm), drizzle with olive oil, bake for about 20 minutes tunring, and then call them special chips. I guess you could make them more chip-shaped if it would help, as long as they're not too thick.
Home-made fish fingers- get something like salmon (or anything really), cut into strips and do the flour, egg, breadcrumbs stuff.
This may have nothing to do with what you were asking about really, and apologies if I'm stating the obvious!