No person is allowed to have friends of the opposite gender. Is it because of trust issues? Yes. Allah does not trust us. In fact, he does not trust us so much (especially in this matter) he doesn't allow us to go anywhere near Zina:
Do not even go close to Zina. It is indeed an indecency and an evil way. (17:32)
Having 'friends' of the opposite gender is definitely not allowed due to the above ayah and other ayah from the Quran. Please note, we are allowed to speak to the opposite gender while keeping a respectful distance and language in a public setting. Online conversation are not in a public setting and are thhus not allowed.
The husband completely has the right to demand his wife not speak privately online with other males, as does the wife have a right to demand the husband not speak to females.