It depends some on your dog, but I wouldn't count on a hedge to keep a dog in. At the least it will need a wire mesh fence in the inside of the hedge to keep the dog contained until the plants grow together. Even a well grown, mature hedge may have some gaps near the base which a dog could slide through, so it may continue to need a wire fence for the first two or three feet.

This request may be a bit of a problem: "shaped like a box (not spherical and not pyramidal)" since in order to maintain dense foliage near the bottom of a hedge, you will need to taper it some so that the upper parts of the hedge don't shade out the lower parts of the hedge. To maintain a boxy tapered shape rather than a more rounded shape will require pruning to give you flat sides.

Do you have deer? If not, there are forms of Thuja occidentalis that will stay shorter and will grow together to make a good hedge. I don't personally have experience with how they take to pruning since I like my plants unpruned for the most part. I have 'Sunkist' (green-gold) which will be easy to maintain in the 6'-10' range for quite some time based on mine as they seem to have slowed considerably at about 5'. I bought plants in the fall of 2007 at about 3'. As individual plants they are a bit floppy in heavy snow/ice situations, but that may be less of an issue with a hedge. 'Techny' is dark green and grows to 10'-15', and I have seen photos of it pruned into a hedge. I planted mine at about 4' in the fall of 2007 and they are about 6' now. I imagine that you can find other selections of this native evergreen that are dwarf in size compared to the regular species (which grows to around 40'.) If you have deer issues they aren't a good choice, but in my heavily wooded rural area, they haven't been bothered.
spring 2010

spring 2014

Another possible choice to look into is a really hardy Rhododendron, though they will have a more rounded form and IMO don't look great pruned into boxy shapes and will make a more gently flowing shaped hedge. They will have the added benefit of spring flowers if you don't prune them. In really cold weather they don't provide a lot of sight privacy since the leaves tend to curl up, but most folks don't require a lot of privacy when the temperatures are below 15 degrees. The link below will take you to a search page from the American Rhododendron Society which allows you to put in height preferences.
On left side of photoPJM rhododendrons in 3/4 shade (so not as dense as full sun,) about 5' at 6 or 7 years in the ground from >1' when planted.

You could ask this question on the shrub forum or the conifer forum for other suggestions beyond what you get here. It also might be worth visiting local nurseries (not big box stores which often have poor or inaccurate info and plants that may not be hardy here) to see what they have in stock that would fit your requirements. They could most probably order more to complete your hedge as it will take a rather large number of plants.

Here is a link that might be useful: Rhododendron Plant Search

You say you are zone 6 NH...which means you must be near the coast? You **might** try a skip laurel hedge. It is an evergreen with large glossy green leaves. It is also a fast grower and can be trimmed into a hedge form. And it is totally deer resistant. My only hesitation is your NH location. Even though it might be officially zone 6, winter is harsher up there, even on the coast, than down here on the far south coast of MA. I grow skip laurels here and they have proven to be perfectly hardy. But I do observe minor burning after harsh winters like last year. So I'm not sure how robust it might be up there. You might try one as a test.

You could try Euonymus kiautschovicus 'Manhattan' grown along a poultry wire fence. It's a very quick grower, not at all invasive (it does not self-sow), and somewhat lax. By poultry wire, I mean that heavy sort of garden fencing that comes in rolls; it has rectangular openings (not like chicken wire, which bends easily), You can use steel stakes or staple it to wooden posts. The wire would keep the dog in, and would disappear behind the Euonymus within a year or so.

I've got a lot of this plant on the perimeter of my yard, and it provides great screening. In some places it's really tall, as it uses other plants for support (I see some near the top of a mature spruce, but see no ill effects on that tree).

When I've mentioned this plant on other threads, some folks have said they wouldn't grow it because it's pollinated by flies. I don't think I have any more flies than normal, but the euonymus is definitely abuzz when it's in bloom. It has nice white flowers and red berries that the birds love.

The pics at the bottom of the page on the Mo. Botanic Garden site are good; there are lots more at other sites.

Here is a link that might be useful: Manhattan at MoBot