There are no official standards by which to distinguish a beautiful problem from a poor one, and judgement varies from individual to individual as well as from generation to generation, but modern taste generally recognizes the following elements as being important if a problem is to be regarded as beautiful:
There are several possible classifications of squeezes, depending on the position. Most common ones are named, some involving a combination of factors:

Mehedinti position
Position after 22. Qf6+