crocodile
There ain't no such animal. There's an old story that crocodiles do this, but it's only a story.
When feeling sad, the lacrimal gland in the outer corner of the eye typically sheds tears first.
Crocodiles are known to shed tears when eating large meals, not from emotion, but because near-choking causes them to produce tears. Other organisms that are capable of shedding tears are elephants and humans.
elephant
eating each other
Both types of men are near to the heart. The man that sheds his tears without feeling he is weak is a true man and a man that sheds his tears, but wipes them away may feel shy or feels that a man should not be crying. For years young boys were brought up not to cry as it was a sign of weakness. Now, it is perfectly acceptable for a man to choose his place to cry if his heart is hurt; he has lost a loved one to death and for many other reasons.
Yes, any animal can.
The most common one is "crocodile tears" which means fake tears; a crocodile is supposed to cry while eating.
canines
Many Tears animal rescue is based in the South of Wales in the small town of Cefneithin. It has animals in several shelters all across the United Kingdom though.
The idiom "crocodile tears" refers to insincere or fake displays of sadness or emotion. It comes from the myth that crocodiles shed tears while eating their prey, appearing to show remorse when in reality they are not genuine.
Dogs, Hogs, Pigs, Cats ETC.