Can I put this in the box, put it into the box, or both?
Both.
Occasionally in or into would make a difference, with in meanside ‘within’ and and 'into meaning ‘from outside to inside:’ ‘He ran in London’ vs ‘he ran into London’ or ‘they bumped in John’ (not very likely, but something could be bumping around side John) or ‘they bumped into John,’ but with ‘put’ either’s fine.
Not both, but either. 
Either option is correct, and both options are correct; both of these statements are correct.
Both.
Both options are correct, but you can’t put this in the box and put it into the box unless you have two of them! (Or you take it out again first.)
Or, you could look at it another way. Either option is correct, but both options would be redundant!
Either way.
In can be used either locationally or directionally: “I slee in my house” vs. “He walked in the house and closed the door behind him.” In the latter usage only it’s synonymous with “into”. (Note in passing that Latin made this distinction by having the object of Latin in be accusative (when it ment into) or ablative/locative (when it meant within).) Connotatively, “into” stresses the idea of entrance while directional “in” takes a broader focus of the full sequence of relocation.
I recall something like this from high school German, where a preposition like “in” takes the accusative or the dative case depending on whether it’s used directionally or locationally.
The verb itself in the OP, put, has an interesting restriction in its modern, non-phrasal use: it always takes an object and a locative expression:
EX:
put this in the box
but never…
*put in the box
*put this
OR:
Could you put the dish on the shelf?
but never…
*Could you put the dish?
*Could you put on the shelf?
You may slee in your house, Polycarp, but you certainly wouldn’t be allowed to in mine. Heck, I make people take their shoes off at the door.
I think the P eloped with Ralph’s missing U’s.