Why do we use square in measuring a qubit with probability? A superposition is as follow:
$$ \vert\psi\vert = \alpha\vert 0\rangle + \beta\vert 1\rangle. $$

When we measure a qubit we get either the result 0, with probability $\vert \alpha\vert^{2},$ or the result 1, with probability $\vert \beta\vert^{2}.$
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Why does the $\alpha$ have the power $2$ ? 

The reason amplitudes are used, which square to probabilities, instead of using probabilities themselves, is that it lets you have the same probability values with different phases.
This is how we can have both of these amplitude vectors represent a 50% chance of a qubit being true or false:
$$1/\sqrt{2}(|0\rangle+|1\rangle)$$
$$1/\sqrt{2}(|0\rangle-|1\rangle)$$
Which is written like this when not using the ket notation:
$$[1/\sqrt{2}, 1/\sqrt{2}]$$
$$[1/\sqrt{2}, -1/\sqrt{2}]$$
Why that is important is because it lets us change the phase without affecting probability.  When we combine values, depending on phase, they will either add together, or cancel each other out.
Here they add together:
$$[1/\sqrt{2}, 1/\sqrt{2}] + [1/\sqrt{2}, 1/\sqrt{2}] = [\sqrt{2}, \sqrt{2}]$$
And here they cancel out:
$$[1/\sqrt{2}, 1/\sqrt{2}] + [1/\sqrt{2}, -1/\sqrt{2}] = [\sqrt{2}, 0]$$
This allows deconstructive interference to happen, which is observable in the real world with experimentation, but is also one of the things that makes quantum computing powerful.
