我可不觉得 - agree or disagree?
The same as "I don't think so".

我 **可**(means emphasize) 不觉得
我不觉得
不觉得
不


When it's a single word "可不", as you post, it is an interjection often used when spoken; it means "exactly" or "you're right".

For example:

你真帅
那可不


And there won't be any usage of this meaning in the form of "我" + "可不",  especially when they are two characters behind "可不". 
Just remember the usage scenarios of oral language like this (a sentence only has 3 characters).

And why can this negative word be used to represent approval?
This is called 反诘 ("rhetorical question").