I do not like that dog sentence, so I will start with Shady_arc comment
"Я не стану учиться на каникулах" (I won't study during my holidays)
It's a wish, proposal for future - but live is hard, parents may think different about that matter, and everything may turn out different, as it planned.
So sentence is expression of who speaks, expression of his current feelings.
But feeling, as any knows, are object of change, especially in long run therms. And highly depends on who makes that statement.
So focus is on the person. And in literature, I wouldn't be surprised, if everything turns to situation where person have to make choose, or do something he denies(even as unlikely possibility) before, especially when it goes about Indian prince's.
So person on first place and unlikely situation in far not defined future (as person thinks)
Also it may be(or actually is) form of oath, vow - in form where unwanted action is not directly connected to the person.(as image in someone else's mind, or it's own mind)
Я не буду есть свою собаку.
Dima's equivalent have different flavor, it's more like decision or promise.
Hard future is now, or can be observed as doom, days, weeks, months.
But anyway here should be a reason, why that question considered to be answered, reasons may be clear from context. One possible reason because you already promised your dog: "Я не буду тебя есть, Пушок, но будет лучше если ты нам нанюхаешь еды." (and really, most times there is much better ways to use your dog to get some food, then just eat that dog.)
Also there is another reason as example - because you do not think that dog meat is tasty. ("свою собаку" - makes that reason a bit unlikely, but)
Я не съем свою собаку
Maybe a promise, but one word ruins all glory : "Я не съем свою собаку, один", "Я не съем свою собаку, один, за раз"
So flavors of considering to eat that dog, but something ruins the plan.
«Ерунда! – воскликнул принц. – Я не стану есть свою собаку!
Говорит нам о том, что человек легкомысленный, не слишком далекий,
но возможно хороший человек, имеющий привязанности
Настоящий младший принц, без забот и хлопот и проблем ))