Not necessarily. Why not start with a more general equation

(*) ax + by + c = 0

>Since
>any vertical line has all points with the same "x" value,
>your slope slope has a denominator of zero.

There's no need to say that. If one can't divide by 0, how is it possible for a denominator to be 0?

Say, find the equation of the line with points (2,1) and (2,3) lie on on it.

2a + b + c = 0
2a + 3b + c = 0

Therefore b = 0, while a could be taken to be 1 and c -2. The equation becomes

x - 2 = 0

No reason to mention the slope at all.

>The slope,
>therefore, does not exist.

Right.

>As a teacher, my advice to all students learning to write
>the equations of vertical and horizontal lines is to accept
>them as they are: special cases that deserve special
>attention (like learning idioms in a foreign language).

It could be useful to mention that "being vertical" is nothing but "being parallel to the y-axis" and similarly for "being horizontal". The moment the plane is rotated even a little bit, the vertical lines cease to be exceptions. Exceptions arise not because lines are vertical or not, by because the coordinate system is usually so chosen as to be formed by a vertical and a horizontal line.