Proof that circuit design problem is NP-hard
If all clauses have the form $y_i \lor y_j \lor (y_k \oplus y_l)$, your problem is in P: simply setting all of the $y_i$ to 1 will satisfy all of the clauses, always.  Therefore, it's unlikely to be NP-complete, and you're unlikely to find a reduction from 3SAT (well, unless P=NP).

But if you additionally allow some clauses to take the form $y_i \oplus y_j$, then your problem becomes NP-complete.  This follows from Schaefer's dichotomy theorem plus a reduction to double the number of inputs (to enable one to express negated literals); or you can find a direct reduction.