There’s two people. Person A eats donuts for breakfast, candy for lunch and cookies for supper to make 2,000 calories. Person B eats fruit, vegetables, whole grains, fish, and lots of oil and nuts to equal 4,000 per day.
Who would be healthier?
You forgot the beer. 
When I was young we were so poor we ate beans for breakfast, water for lunch, and swell up for dinner. No shit.
Person B, no contest. All that crap does really bad things to your insulin sensitivity.
You forgot “all else being equal”. Which I assume is unlikely, as person B will be full of energy and ready to exercise.
Assuming “all else being equal” means no one is taking any vitamins, person A could have some near term problems with vitamin deficiency, like scurvy. I’d say this depends mostly on how extreme the junk food only restriction is on person A. If they have even vitamin-fortified sugar coated cereal, it could make a difference for them. But if it is literally no plant food except white flour, it seems like they could have malnutrition within less than a year.
Person B could be gaining 4 pounds a week on that 4,000 calorie diet (assuming required maint. calories =2000). In the short term that won’t matter, but after a year when the person is carrying an extra 200 pounds or so it will cause some health problems.
I’ve been doing a lot of (admittedly informal) research into nutrition ever since starting down the weight-loss road a couple months back, and from everything I’ve read, this is my simplistic understanding:
Calories are the only thing that determine whether you gain or lose weight. Doesn’t matter what you eat, if you eat more than you burn in a day, you gain, and if you eat less, you lose.
However, strictly aside from weight control, there’s nutrition to consider. I could eat 1500 calories worth of Skittles and lose weight, but that diet lacks protein, vitamins, minerals, healthy carbs, etc. etc. You’d lose weight, but suffer from malnutrition.
Presuming the same amount of exercise for each, person A would be skinny but ill/prone to illness, while person B would be fat but relatively healthy.
When I eat like person A, I want to sit down and watch TV all day. When I eat like person B, I want to go play racquetball, tennis, ride my bike, or something. My vote is for person B.
No shit? Meaning, you couldn’t poop, because you couldn’t pass the uncooked beans through your system? 
It would be a ‘political shit’: all wind and piss and very little substance. 