To the best of my knowledge, there is no such thing as an equivalent to “cooties” in French “childhood culture”, either in Europe or North America. You might find references to the other sex being icky, but I have never seen it “formalized” as something like it is in English.

I’m French.
The best translation I can think of is (as, apparently, “cooties” is an insult boys use against girls) “les filles c’est nul” or “les filles ça craint” (not “ça craint = it scares” but “ça craint = it’s the shame of being with, it’s boring”. Literary translation would be “les filles c’est des pouilleuses”, “the girls are dirty / verminous persons”, but it is not something children say in real life. This is too “nasty” and saying that would probably result in the children being scolded by their parents; saying “les filles c’est nul” might make parents smile.

In French, we would use something like “la touche pas, elle a des poux” (don’t touch her, she has lice).

Je ne connais pas l’expression anglaise, mais en France quand j’étais gamin j’entendais souvent les garçons moquer les filles en les qualifiant de pisseuses.

A possible translation for “cooties” (which works for speakers from Québec or Europe) would be « des microbes de fille ».