Will an American marrying a Slovenian give him US citizenship and her EU citizenship?

No on both accounts. Marriage would only increase the likelyhood that he would later on obtain citizenship, and with you, you would actually have to be in Slovenia for x number of years, or if ever to obtain legality in that country. It's a long drawn out process on both sides. Here in the states, it has become an expensive process as well. Go check out the immigration laws on the internet. You will soon find that it is not all that fun.