How to determine u(t+1) - u(t-1) as even, odd or neither?
The technique to determine if it is even or odd (or neither) can be either graphical or analytical, depending on what is simpler to do. That from a practical point of view.

If you are a student and you are required to prove your conclusions, the acceptable way depends on which rules are set by your instructor.

Anyway, the analitical conditions to be satisfied are:

$$
  x(t) \mathrm{~is~even} \qquad\Leftrightarrow\qquad \forall t \qquad x(-t) = x(t)
$$

$$
  x(t) \mathrm{~is~odd} \qquad\Leftrightarrow\qquad \forall t \qquad x(-t) = -x(t)
$$

From a graphical POV the first condition means that the graph of \$x(t)\$ is symmetrical about the vertical axis, whereas the second condition means that the graph is symmetrical about the origin.

Since this seems some sort of homework question, I won't give you the full solution (this is the policy of this site wrt. homework questions).