He needs some type of visa or entry document to return to the U.S. You can also try to apply for a refugee travel document. If he returned to the country from which he is an asylee this may be very problematic.
The above is intended only as general information, and does not constitute legal advice. You must speak with an attorney to discuss your individual case.
No, it is highly unlikely that he will be let back in. He has no admission. You will need to file a separate petition to bring him back as a derivative asylee.
Behar Intl. Counsel 619.234.5962 Kindly be advised that the answer above is only general in nature cannot be construed as legal advice, given that not enough facts are known. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers will require cognizance of all pertinent facts about your case. Any answers offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship.
Did your husband already have his green card? If he has his green card that is all he will need to reenter the USA. As a derivative asylee, there is no issue with him returning to his home country - that would be a problem for you as the principal.