Does 'Swiss' in that sentence means people from Switzerland? if yes, the correct answer is the 2nd sentence.

I know he speaks French, German and Italian so he can't be Swiss. (Incorrect - this sentence, whilst grammatically correct is not true as many Swiss citizens do speak these languages; but also a person speaking those three languages could be from anywhere, including Switzerland). I know he speaks French, German and Italian so he could be Swiss. (Correct)

In terms on syntax, those sentences are both correct in the "subjunctive", however "subjunctive" is hardly ever used, in spoken English Therefore, if you were wanting to convey such a meaning, such as you have, apparently intended, in spoken English, you should say: "I know THaT" ie "I know that he speaks French.........." Grammatically, you have made one small error: you need to place a comma ",", after Italian ie "I know that he speaks French, German and Italian, so he can't be Swiss".