Motivation:
eq and ne should be faster.

I don't think that this is the case when comparing AnyRef to a literal null, at least not in any in modern Scala versions.

Consider:

class C {
  def f(x: AnyRef): Boolean = (x == null)
  def g(x: AnyRef): Boolean = (x eq null)
}

In Scala 2.13.2, via javap -c -v target/scala-2.13/classes/C.class, these compile to identical bytecode where both use the ifnonnull instruction:

public boolean f(java.lang.Object);
    descriptor: (Ljava/lang/Object;)Z
    flags: (0x0001) ACC_PUBLIC
    Code:
      stack=1, locals=2, args_size=2
         0: aload_1
         1: ifnonnull     8
         4: iconst_1
         5: goto          9
         8: iconst_0
         9: ireturn
[...]

public boolean g(java.lang.Object);
    descriptor: (Ljava/lang/Object;)Z
    flags: (0x0001) ACC_PUBLIC
    Code:
      stack=1, locals=2, args_size=2
         0: aload_1
         1: ifnonnull     8
         4: iconst_1
         5: goto          9
         8: iconst_0
         9: ireturn
[...]

Similarly, I also don't see a difference in bytecode for the ne case. I also don't see any bytecode differences in Scala 2.12, Scala 3, etc.

Edit: to be clear, eq and ne could potentially make a performance difference in cases where you're doing more than a null check and only need reference equality, but I don't think it makes a difference for null checks.