I am trying to grow some squash at home.  Our most promising squash, a
winter melon, has long thick vines.  One of these vines cracked
perpendicular to the vine's growth such that the vine was nearly split
in half.  I tied the broken edges together and put some support
underneath the vine so that there wasn't stress on the injury.  Is it
possible for the vine to recover?  Can plants heal from injuries the
same way animals do?  Are there any hormones or fertilizers that would
encourage it to repair itself?

If not, what is the best solution for fixing the plant so the unbroken
parts grow well?  One potential problem is we cut off the side
branches that the plant grew so that it would focus its energy on the
main branch.  Now the end of the main branch is damaged in this
manner.  Will the plant eventually grow new side branches if the
growing point of the main branch is dead?

Sounds like SQUASH VINE BORERS
http://hgic.clemson.edu/factsheets/HGIC2207.htm

SQUASH VINE BORERS

The squash vine borer (Melittia cucurbitae)
http://hgic.clemson.edu/factsheets/Graphics/cucurbins/borer.htm

ranges from Canada to Argentina and is the most serious enemy of
squashes and gourds. It causes much trouble where only a few plants
are grown in gardens. It rarely attacks cucumbers and melons. Great
variations exist in the susceptibility of squash and pumpkin
varieties. Butternut and Green-Striped Cushaw varieties are
practically immune to attack, but Hubbard squash is highly
susceptible.

Damage is caused by larvae (immature forms) tunneling into stems. This
tunneling often kills plants, especially when the larvae feed in the
basal portions of vines. Sometimes fruits are also attacked. Sudden
wilting of a vine and sawdust-like insect waste coming from holes in
the stem are evidence of attack.

The adult is one of the moths known as "clear wings" because the hind
wings are almost without scales. It is 1½ inches (3.8 cm) in wing
expanse and metallic greenish black in color. Hind legs are fringed
with black and orange hairs, and markings of similar color occur over
much of the abdomen.

The moths are day fliers and are often mistaken for wasps. Larvae are
white, heavy-bodied and considerably over 1 inch (2.54 cm) long when
fully grown.

The insect overwinters in the soil as a larva or pupa (a nonfeeding
stage where the larva changes to an adult) enclosed in a cocoon. Moths
emerge in early summer and lay eggs on the stems of the plants,
usually late May in the South. Upon hatching, larvae bore into vines
and complete their development in four or more weeks. Then they leave
the plant, crawl into the soil, spin a cocoon and transform to a pupa.
There are two generations in South Carolina.

In a vegetable garden, various measures can be taken to control this
pest. Till the soil in late winter to expose overwintering insects.
Rotate squash to another location in the garden each season. Destroy
vines that have been killed to break the life cycle. You can slit the
infested vine lengthwise and remove borers or kill them with a long
pin or needle. Place soil over slit stem after removing the borer to
encourage root development, and keep plants well watered. Plant as
early as the weather allows since borers do not emerge until early
summer.

I should have mentioned the crack occurred when we were moving the
plants.  Thank you for the warning about these pests, though.

The crack occurred near the end of the vine, so there are plenty of
leaves that can absorb sunlight.  My concern is that the vine won't be
able to grow any further if the growing tip dies.  We already trimmed
the side branches to force the plant to grow in one direction.  Is
there anything I can do to help that part of the plant survive this
injury?

I don't have anywhere to post a picture, so I will post an ascii drawing:

   --------------------------------
     this way                 |
 <-- to roots     -------------      --> this way to growing end
                     crack
   --------------------------------

As I mentioned, the crack occurred when moving.  We tied string around
the broken part so that the injured surfaces were touching each other.
 Is there anything else we can do to facilitate healing?

Rather than using string, you can buy green vinyl tape for tying up
plants that won't cut or damage plants.

Will the crack heal if I do that?  If it won't heal back into one
piece, but I keep the tape in place, will the part of the plant on the
other side of the crack continue to get nutrition and continue to
grow?

Using gardner's tape is not a matter of healing the cut area, but more
to do with not damaging the stem/vine with string - tape is generally
used to reduce potential harm and is very strong.  It's very
inexpensive.

Regarding nutrition/healing on the cracked side, I really don't have
an answer for you.  I also don't know what you could do to facilitate
healing.  If you call your local nursery, or county extension office,
possibly they'll be able to advise you.

Without seeing it, I couldn't give you a good answer.  Sorry!

This type of tape is used by florists every day. Stop in to your local
flower shop and offer them a buck for a roll of tape. This tape is
green and very thin, it stretches a bit, and it's a bit pourous which
allows the stem to breathe. I bet it works.

Yes, that is what I had understood before the picture.  I think TLS
was pointing to the part about splitting the vine lengthwise, but that
is, of course, a much less damaging operation, hence the
recommendation.
I am absolutely no expert, but I would say that the only thing you can
do is the protect the break, maybe with a splint if it is likely to be
moved again, and try the keep it out of the sun and from drying out. 
I don't think it will grow back, so I'd expect that only half the
nourishment will get past the wound.  Sorry.

(I was going to suggest a tetanus shot and mercuricome, but you may
feel this is no laughing matter.)
Myoarin