¿Cuál es la diferencia entre We all have a right to be happy. y We all have the right to be happy. ?Puedes dar oraciones como ejemplo.

Nothing really. Using "a" simply puts an emphasis on the notion that this right is one of many. Using "the" puts the emphasis on the particular right you are talking about (happiness).

It's probably safer to say "a right" in the same way one would say:
We all have a pencil.
instead of
We all have the pencil. <--- wrong.

The reason "the right" works in your example is because it will be interpreted as we all sharing the same idea. Many people can share an individual concept (a right to be happy) but can't share certain physical things at the same time (like a pencil).