The Patronus charm is incredibly personal to each wizard who conjures it. It's so personal in fact that, unlike an animagus form or Hogwarts house, it can change with the personality of the conjurer. Dobby and Ignotus Peverel (the youngest brother in the "Tale of the Three Brothers" and ancestor of Harry Potter) are both Cancer within the Harry Potter universe.
Likely - Crab
The Crab is the very sign of Cancer and thus would make a very appropriate patronus for any wizard born under the Cancer zodiac. Crabs are known for being able to live both deep in the ocean and on the beach. Like any good Cancer they are adaptable. Quiet and observant (consider how a crab scuttle away when uncovered in the sand) they are also, like Cancer, very protective (that's what their claws are for, of course). Crabs can also be easily hurt, due to their size which, again, makes them a perfect match for Cancer.
Not - Goat
Goats are known for their ability to survive. They are also not terribly friendly (they bite and spit). Goats don't care deeply the way Cancer do. It also doesn't help that the goat is a symbol of Capricorn, the opposite sign from Cancer on the zodiac. The odds of a Cancer wizard producing a goat as their corporeal patronus are slim. Mountain goats, in particular, are also very sturdy, so they are very hard to hurt.
Likely - Wolf
The moon is the ruling planet of any Cancer, and thus a wolf would make sense as a patronus of a Cancer. A wolf would be particularly impressive when taking on dementors and would certainly increase the confidence of the wizard it represented.
Not - Crow
Crows are not particularly adaptable, as Cancer are known to be. They don't leave their comfort zones very often. They are also known for being a bit demanding, rather than adaptable and sensitive.
The ambition of crows is also a trait that fits other zodiac signs much more than it does with Cancer. Crows are too smart to be the most dependable of creatures. It's not a bad thing, just a difference.
Likely - Seal
Being caring and protective can make Cancer very playful, very much like a seal. The seal can hang out on the surface of the water or dive deep under water, so they, like Cancer, are also very adaptable. Emotionally Cancer can also dive deep, it's part of the reason they are often seen as moody, they are just introspective and distracted. Cancer, like Pisces, is a water sign, and thus a seal matches up on another level as well.
Not - Elephant
Elephants are not moody, they have a lot of self control, so there's a difference right away. Elephants forge ahead, but due to their size aren't terribly adaptable to different environments or situations, a key trait of any Cancer. As the largest land mammal it's also difficult to injure an elephant, where as a Cancer is sensitive and easily hurt very often. While an elephant would make a very impressive corporeal patronus it's unlikely that a Cancer wizard would be the one to conjure such a shape to represent them.
Likely - Dolphin
Like the crab or Cancer itself, the dolphin, as an animal that lives in the ocean, fits nicely with Cancer as it is a water sign. Like Cancer, dolphins are very protective, especially of their families. They are also known for interacting and sometimes rescuing humans from trouble out in the ocean. They are sensitive, and able to communicate and relate to others. They are also adaptable, as they swim deep underwater as well as enjoy time at the surface.
Not - Crocodile
The crocodile is known for being particularly ambitious, focusing, and sitting for long periods of time waiting for prey to get near. This focus and self control doesn't match Cancer terribly well.
They are not particularly sensitive or easy to hurt, again pushing them away from being a great representation for Cancer. They may seem moody, but that's not quite enough to make them the best representation for the sign of the crab.
Likely - Sea Lion
Like the seal and the crab before it, the sea lion is adaptable, hanging out both under water and in the air. They are also a great representation for any water sign, of which Cancer is one. They live in the ocean and very much enjoy their beach time as well. Sea Lions sacrifice themselves to protect their children, and are known for their loyalty and attachment to mates, family, and trainers.
Not - Goose
Looking for an animal that's mean? The goose is right there ready to attack ers by. It's not easy to predict their behavior. They may float on water and fly in the sky, but they're not particularly adaptable, or they wouldn't have to flee the north every time a chill enters the air. They can be very patient and focused, until they go off. Goslings can depend on their parents, but no one else can really depend on a goose.
