Grammatically, never creates a negative. This is important to remember because you should avoid double negatives while writing:
Wrong: I never received nothing.
Right: I never received anything.
Wrong: Nothing never changes.
Right: Nothing ever changes.
Wrong: I have never heard nothing so clearly.
Right: I have never heard anything so clearly.
While some words have a positive connotation (such as fantastic) or a negative connotation (such as loser), this isn't the case with never. It just doesn't make sense to look at it like that.
Fortunately, it's usually easy to determine what it means:
She never said that.
This means she did not say that, at any point. (You will know what "that" is from context, of course.)
Never run with scissors.
This is a command (using the imperative). It's the same as saying "Do not run with scissors".
Your two examples, on the other hand, use never for emphasis:
I have never been happier.
It has never been better.
In the first example you're really happy, and in the second example something's going well.
You can tell never is being used for emphasis because of the implied comparison. When you find a sentence with never and an implied comparison, make the comparison explicit and parse like normal.
So your one example becomes "I have never been happier than [how happy] I am now". (Of course, the comparison varies depending on what tense is used. "I had never been happier than [how happy] I was then" is how it would be in the past tense.)
It should be easier to understand now.
Both of your examples have a comparative word (happier and better), but there are numerous other variations on how this type of sentence can be formulated. Although I'm trying my best to cover as many as possible, there are likely more out there.
Your examples can be rewritten with so:
I have never been so happy.
It has never been so good.
If you look back, you'll see that "I have never heard anything so clearly" also works this way.
You can also use inversion:
Never have I been happier.
Never has it been better.
For some reason you can also say "I have never been more happy", but not "never been more good". (The best explanation I have for this is "because English".)