Can == be used on enum?

Yes: enums have tight instance controls that allows you to use == to compare instances. Here's the guarantee provided by the language specification:

JLS 8.9 Enums

An enum type has no instances other than those defined by its enum constants.

It is a compile-time error to attempt to explicitly instantiate an enum type. The final clone method in Enum ensures that enum constants can never be cloned, and the special treatment by the serialization mechanism ensures that duplicate instances are never created as a result of deserialization. Reflective instantiation of enum types is prohibited. Together, these four things ensure that no instances of an enum type exist beyond those defined by the enum constants.

Because there is only one instance of each enum constant, it is permissible to use the ==operator in place of the equals method when comparing two object references if it is known that at least one of them refers to an enum constant. (The equals method in Enumis a final method that merely invokes super.equals on its argument and returns the result, thus performing an identity comparison.)

This guarantee is strong enough that Josh Bloch recommends, that if you insist on using the singleton pattern, the best way to implement it is to use a single-element enum (see: Effective Java 2nd Edition, Item 3: Enforce the singleton property with a private constructor or an enum type; also Thread safety in Singleton)

What are the differences between == and equals?

As a reminder, it needs to be said that generally, == is NOT a viable alternative to equals. When it is, however (such as with enum), there are two important differences to consider:

== never throws NullPointerException

  enum Color { BLACK, WHITE };
  
Color nothing = null;
  if (nothing == Color.BLACK);      // runs fine
  if (nothing.equals(Color.BLACK)); // throws NullPointerException
  

== is subject to type compatibility check at compile time

  enum Color { BLACK, WHITE };
  enum Chiral { LEFT, RIGHT };
  
if (Color.BLACK.equals(Chiral.LEFT)); // compiles fine
  if (Color.BLACK == Chiral.LEFT);      // DOESN'T COMPILE!!! Incompatible types!
  

Should == be used when applicable?

Bloch specifically mentions that immutable classes that have proper control over their instances can guarantee to their clients that == is usable. enum is specifically mentioned to exemplify.

Item 1: Consider static factory methods instead of constructors

[...] it allows an immutable class to make the guarantee that no two equal instances exist:a.equals(b) if and only if a==b. If a class makes this guarantee, then its clients can use the ==operator instead of the equals(Object) method, which may result in improved performance. Enum types provide this guarantee.

To summarize, the arguments for using == on enum are:

• It works.
• It's faster.
• It's safer at run-time.
• It's safer at compile-time.