Why is the minus sign (-) not allowed in reaction equations?
Don't get too caught up in notational rigidity. You're "allowed" to use any notation you want, if it gets you to the correct answer. If you really want, go ahead and use $\ce{Na\ - \  e^{-} -&gt; Na+}$, or go all out and use $\ce{-Na^{+}\ - \  e^{-} -&gt; - Na}$. However, as Richard Feynman discovered while learning trigonometry, if you start using too much obscure/uncommon/personal notation, you get into trouble showing others your work, which makes doing science hard. Standardizing notation is just a way to save time and headaches later on.

The most basic chemical equations actually carry very little chemical information, to the point that it's easy to write reactions which are chemically outrageous. One of their most fundamental purposes is simply to establish atom conservation (which is why so much time is spent on teaching how to "balance" reactions). Thus, chemical equations are always equivalent to a system of linear equations, which is why you can treat them like such and add/subtract molecules to both sides, multiply by a number, add two different reactions together, etc. Using negative stoichiometric coefficients would thus be a very trivial extension of this behaviour, and the equation would remain completely valid. 

Let's look at an example. Take the intermolecular dehydration of ethanol to produce diethyl ether:

$$\ce{2 C2H5OH -&gt; C2H5OC2H5 + H2O}$$

The equation is just as valid if you subtract a water molecule from each side:

$$\ce{2 C2H5OH\ -\ H2O -&gt; C2H5OC2H5}$$

Not many people would write the above equation. However, we do have another type of standardized notation which is often used, and which involves a negative stoichiometric coefficient. Using this notation, the equation is written as:

$$\ce{2 C2H5OH -&gt;[\ce{- H2O}] C2H5OC2H5}$$

The water molecule has a legitimate negative stoichiometric coefficient of -1, and again the equation is exactly as valid as all of the ones above. Why is it that negative stoichiometric coefficients are rarely written in the reactants or products, but common when written atop the reaction arrow? I have no idea, but ultimately it makes no difference. That's just how notation happened to become standardized, and if everyone understands it, that's fine.