This article suggests that originally the expression was: "parler français comme un Basque espagnol".
Is this expression correct in English?
It can't be correct or not since it's a proverb. Proverbs may not be very grammatical.
Is there a similar expression which refers to non-native speakers who don't speak the language well, or who have a very pronounced accent?
I'm not aware about direct proverb counterpart in English, so I would suggest translating it by its meaning:
"To speak broken English" — incorrect or awkwardly structured English, usually spoken or written by non-native speakers (Urban Dictionary)
TL;DR: The problem with translating proverbs is need to convey the context. Historically, people are only aware about neighboring nations, and, of course, there are jokes about each other. An average English (or American) reader or listener may not be simply aware what's wrong with Spanish cows (or Spanish Basques). It has changed in a recent centuries, but still not sufficient to understand tiny details of a humorous context.