You are right to question this.  The product
$\prod_p \left(1 - \chi(p)/p\right)^{-1}$
(where $\chi = (-1/\cdot)$ is the Dirichlet character mod $4$) 
does converge, but that requires justification; indeed it is equivalent to 
the non-vanishing of the Dirichlet function $L(s,\chi)$
on the edge $s = 1+it$ of the critical strip, which is also what you need
to prove the analogue of the Prime Number Theorem for primes in arithmetic
progressions mod $4$.  (Taking logarithms we see that the desired convergence
is equivalent to convergence of the sum $\sum_p \chi(p)/p$, which differs from
the product's logarithm by an absolutely convergent sum $\sum_p O(1/p^2)$,
and from that to $L(s,\chi)$ is classical analytic number theory.)