box

In a restaurant, if you ordered more than you could eat, you could ask the waiter/waitress to bring you a box. Then he/she will box the food for you so that you can take it home and enjoy it later.

Is the use of "box" correct here? Before I came to the United States, I was told that we could use dog-bag. But I went to several restaurants and I have not heard anyone say dog-bag. 

The correct term is 'doggie-bag'. 'Box' in terms of take-out items is used, for example, as"Could you box this up?", but I don't think it is used for restaurant food. Chinese restaurants  sometimes serve food in a box, in which case you might ask for a box. But, I think 'doggie-bag',  even though it is commonly understood, is not in general use anymore. Nowadays people are more likely to ask to 'get something to go' or to 'wrap something up'.

Can it be something with packet, packeting?

Styrofoam flip-top boxes (or hinged-lid boxes) have almost entirely replaced "doggy bags" in my neck of the woods.  In some restaurants the server does it for you (or offers to) and in others it's do-it-yourself.  If you ask for a "doggy bag" everyone will know exactly what you mean, and will bring you a box.  In most cases where the restaurant accepts "to-go" orders, the same boxes are used.

If the server sees that you've finished eating and have left a substantial  amount, he/she may ask, "Would you like a box?", or "May I box that for you?"

About the only places that do this in the UK are pizza restaurants. You'd normally just ask 'can I take this with me?' or something.