Michael Hardy and Catherine Woodgold have written a very nice and readable account on the misconceptions about Euclid's proof (published in 2009). I'm sorry that I can only give a paywalled link; at least the first two pages are available. http://link.springer.com/article/10.1007%2Fs00283-009-9064-8
Incidentally, with the specific situation at hand, the question "constructive vs. nonconstructive" is slightly moot. This is because there is a certain metatheorem in mathematical logic which states: If there is a nonconstructive proof of a statement, then there is also a constructive proof.
Of course this metatheorem doesn't apply to arbitrary statements, only to statements of a specific logical form (so called "geometric sequents"). But the statement "there are infinitely many prime numbers" can be put into such a form. Also, in case you are wondering, this metatheorem admits itself a constructive proof.
Summarizing, there is a mechanical way to turn any nonconstructive proof of the infinitude of the primes into a constructive one.
The key words to look up here are "double-negation translation" and "Friedman's trick". Fantastically, the double-negation translation turns out to be "the same as" the continuation-passing style transformation, if viewed from the right angle. Some pointers are in this slide deck: http://rawgit.com/iblech/talk-constructive-mathematics/maste...