The way I understand it, the New Hampshires (I thought they were Reds too until this thread. I also learned something today) were developed from RIR to be more of a meat bird. The breasts should be broader (more white meat) and the lighter color gives a prettier carcass considering pin feathers compared to the RIR. So they should be a different color and a different body shape.
The original New Hampshires were not known as great egg layers since they were developed as meat birds. Whether or not the New Hampshires you get are great egg layers is going to depend on the strain. What that means is that if the breeder that has been selecting which birds to breed has been selecting for egg production, they might be pretty good. If egg production is not a high priority in selecting the parent birds, then they are probably not great egg producers.
I am not talking about hatchery birds as much as I am talking about birds from breeders that know what they are doing. Hatcheries will sell birds that are sort of the right color and sort of the right body shape, but they are not nearly as selective in ther breeding program as somebody that really follows the Standard Of Perfection. Hatchery birds will often lay well though because that is one of the traits the hatcheries want to encourage in their flocks.
After learning more about this breed in the last few months, they are very high on my radar screen when it comes time to introduce some fresh genes to my flock's gene pool, but my interest is more in meat than eggs.