NP-completeness proof via reduction
Suppose you are given a 0-1-Integer-Programming instance. You need to substitute $x_i$'s, $x_i \in \{0,1\}$, with new variables $y_i$'s as G. Bach mentioned. The substitution is $x_i = y_i/2$, $y_i \in \{0,2\}$. 

By this substitution, you get
 0-1-Integer-Programming $\leq_P$ 0-2-Integer-Programming
 and also 0-2-Integer-Programming $\leq_P$ 0-1-Integer-Programming by reverse substitution.
(This is because the polynomial reduction function in this case is invertible).

Therefore 0-2-Integer-Programming is also NP-complete
(from 0-1-Integer-Programming $\leq_P$ 0-2-Integer-Programming).

By the way, you are not allowed to add $x_i \neq 1$ in canonical integer programming problem. Basically it means $x_i \leq 0$ OR $x_i \geq 2$. We can't have OR constraints, all we can have is AND constraints.