you have a very good question. and the answer is very difficult to me ㅜㅜ
First of all,
you can use the both 니가 보고 싶다 or 네가 보고 싶다.
We use 니 (you) in spoken type ,not written type. So, when you write something, you should write 네(or 너),but when you speak in Korean, 니가 보고 싶다 is common.
Okay, 너를 보고 싶다 is not right. 니가 or 네가 보고 싶다 is right
We usually take off subject( when we know exactly what it is). So, you can say like this,
나는 니가 보고 싶다.( but, we take off the subject " I" because the person who say now is I.),so we know clearly who say now. Therefore, 니가 보고 싶다. but you can say like this.
나는 너를 그리워한다. ㅋㅋㅋ I cannot explain clearly because language is a kind of habbit that we use everytime or long time ago. So, you should memorize that kind of thing. Actually I don't understand why we should use like that even though I'm native Korean. lol. Sorry that I cannot explain clearly.