How to use meditation to boost logical thinking (or avoiding suffering with unrational scenario)

To an extent, yes. Meditation stills the intellect, and expands wisdom. As your wisdom expands and your intellect stills, you will end up closer to experience. This closeness facilitates a sharpness and mental clarity that can make problem solving more natural.

I think that the point of meditation isn't to enable logical thinking. If you let yourself be carried away by the trains of thoughts, it doesn't matter if it makes you feel bad or good. Meditation is to be present and aware.

Is there anyway we can use meditation as a tool to calm ourselves AND show to us in a very effective way that this fear is unrational and this terrible scenario is highly unlikely?

Meditation is more about perceiving that there is no sense to imagine too far, or to get excessively concerned with thoughts. I'm not saying that thinking is bad, but be more pragmatic. Instead of thinking too far away and producing suffering, try to do something practical. Try something like "ok, this scenario don't look very good, what is the simplest thing I can do now to reduce the possibility of it?"

Meditation, I think, is very related to pragmatism. When you meditate, you just meditate and nothing more. And bringing it to life, instead of becoming overwhelmed by the possibilities, try to find what you can do right here and right now that can contribute to avoid the bad cases.

It is a change of focus. Instead of feeding the image of some possible future, come up with something that can be done immediately. And if you can't do anything about it, let it be.

Again, instead of improving your thinking or changing the feelings that thought produces, meditate to be more pragmatic, more present and more aware.

And lastly, a book recommendation. I think that being pragmatic is one important point in Shunryu Suzuki's "Zen Mind, Beginner's Mind".

Seated mediation, as I've been taught to practice, is not about logical analysis. That said, when it comes to 'thinking the worst', the practice of attention recognizes that these are just thoughts. In a sense, it doesn't matter whether the events are likely or unlikely, because the thoughts are just thoughts. There can be value in rationally analyzing fears, but meditative practice starts with just observing the fears as they are, regardless of whether they are rational. When the whirlwind of doubt starts up again, just return to the object of meditation (often, the breath) and let things settle. This isn't easy, but that's why we practice.

A bit of an aside, but here is a passage from the Dhammapada, Mind (translation by Ananda Maitreya), that I think relates to this:

Just as an arrowsmith shapes an arrow to
  perfection with fire,
So does the wise man shape his mind,
Which is fickle, unsteady, vulnerable, and
  erratic.

Like a fish taken from the safety of its watery
  home
And cast upon the dry land
So does this mind flutter, due to the lure of the
  tempter.
Therefore one should leave the dominion of
  Mara.

How good it is to rein the mind,
which is unruly, capricious, rushing wherever
  it pleases.
The mind so harnessed will bring one happiness.

A wise man should pay attention to his mind,
Which is very difficult to perceive.
It is extremely subtle and wanders wherever it
  pleases.
The mind, well-guarded and controlled,
Will bring him happiness.

One who keeps a rein on the wandering mind,
Which strays far and wide, alone, bodiless,
Will be freed from the tyranny of the tempter.

A man of fickle mind
Will never attain wisdom to its fullest,
Since he is ignorant of the Dhamma
And has wavering faith.

The heart of the fully conscious man
  is fearless--
He has freed his mind of lust and anger,
He has transcended both good and evil.

Observe this body, as fragile as an earthen vase.
Build a mind as solid as a fortified city,
Then confront Mara with the weapon of
  insight
And (proceeding without attachment)
Guard what you have already conquered.

Certainly before long this body will lie on the
  ground,
Lifeless and unconscious,
Cast aside like a useless log.

A mind out of control will do more harm
Than two angry men engaged in combat.

A well-directed mind creates more well-being
Than the wholesome actions of parents
Toward their children.