Is a man whose car has broken down and who has no money considered one of those eligible for Zakat, and is it permissible to give him the Zakat amount (20 Kuwaiti Dinars) in this case?

If you give dinars to a man with the intention of zakat, and you predominantly believe he is among those eligible for it, that is sufficient for the zakat to be valid. It is enough to inform him that he is in need. If someone whose wealth is not known claims poverty, his statement is accepted. If it becomes apparent after payment that he was not deserving, it is more cautious to give the zakat again. However, the Hanbalis are of the view that it is valid as long as you believed he was deserving at the time of payment. But if you did not intend it as zakat at the time of payment, or you predominantly believed he was not deserving, then your obligation is not fulfilled by that.