We know that Force equals Mass times Acceleration, $F = ma$. But what if we had a law like $$F=m^2a$$ or $$F = 2ma~?$$
Answer
It wouldn't change anything. We would just have defined mass differently. What is relevant about Newton's 2nd law is not $m$ but rather the fact that $F$ and $a$ are proportional, $F\propto a$.
The proportionality constant can then be called $m$ or $2m$ or $m^2$. That doesn't matter for the relationship that this law describes.
Newton's 2nd law defines mass (inertial mass) as the proportionality constant in $F=ma$. Had we decided upon $F=2ma$, then we would just have defined mass as half of the proportionality constant. It would change our language regarding mass (you would only weigh 40 kg rather than 80 kg i.e.), but it wouldn't change the way the world works.
If you on the other hand had asked about $F=ma^2$ or so, then you are messing with the world entirely with disastrous consequences that we can only try to guess as a result!
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