Only ‘right’ is natural and its meaning is different in the two sentences. I’m behind you. (Somewhere) I’m right behind you. (Close to you) Everything is planned down to the last detail. (Natural and clear) Everything is planned right down to the last detail. (‘right’ barely changes the meaning. It provides a small amount of emphasis.) Natural uses of ‘entirely’. He’s entirely correct. (All aspects of what he said are correct.) It’s entirely inside the box. (No part of it is sticking out)

1. right - immediately behind you. entirely - metaphor, I support you 100% 2. No difference, can use either word. Meaning. completely, total without error or planned to every detail.

This is a good question and I am not a tutor, but in these sentences they mean the same thing. The only exception is that in the first sentence 'right behind' someone can also mean physically behind them in a queue. They would then be immediately behind someone, i.e. the very next person. This is instead of the meaning here, which is being supportive of someone. Also, in Britain, we tend to say 'everything is planned right down to the last detail'. Therefore, 'entirely down to the last detail' sounds a little odd.