Is multiple equation 0/1 linear integer programming with bounded integer weights NP-hard?
This problem is NP-complete, by reduction from 1-in-3 positive 3SAT.

1-in-3 positive 3SAT is a variant of 3SAT, where we require that exactly each clause consists of 3 positive literals (no negation), and to satisfy the formula, exactly one literal in each clause must be true (no more, no less).  Thus, each 1-in-3 positive 3SAT instance is a formula

$$\varphi(x_1,\dots,x_n) = \bigwedge_i (x_{c_{i,1}} \sqcup x_{c_{i,2}} \sqcup x_{c_{i,3}}),$$

where $x \sqcup y \sqcup z$ is true iff exactly one of $x,y,z$ is true.

You can immediately convert this to an instance of your problem.  In particular, the $i$th clause corresponds to an equation

$$x_{c_{i,1}} + x_{c_{i,2}} + x_{c_{i,3}} = 1.$$

For instance, the clause $(x_3 \sqcup x_5 \sqcup x_6)$ corresponds to the equation $x_3 + x_5 + x_6 = 1$.
We get one equation per clause.  The equations are simultaneously satisfiable iff the formula $\varphi$ is satisfiable.  This completes the reduction.