Dirichlet's theorem on primes in arithmetic progression

As Jonas says, Keith Conrad's paper will tell you that the answer is essentially no. He defines a certain notion of "Euclidean" proof coming from writing down a polynomial with certain properties and gives two classic results which say that these proofs exist for primes in arithmetic progression $a \bmod n$ if and only if $a^2 \equiv 1 \bmod n$.

The first case these proofs can't handle is primes congruent to $2 \bmod 5$ (equivalently, primes congruent to $3 \bmod 5$). The basic problem is that there is no way to force a positive integer to have factors congruent to $2 \bmod 5$ and not congruent to $3 \bmod 5$ (or the other way around) solely by controlling its residue class $\bmod 5$. In more sophisticated language, $2$ and $3$ lie in all the same subgroups of $(\mathbb{Z}/5\mathbb{Z})^{\ast}$.

A couple more references:

Ram Murty and Nithum Thain, Prime numbers in certain arithmetic progressions,

Paul Pollack, Hypothesis H and an impossiblity theorem of Ram Murty.