Is my husband eligible for a Green Card?

Too long (and often repeated) story for a general blog such as this. Consult with an experienced attorney in-person or over Skype to learn about your options.

If you are a U.S. resident or U.S. citizen you can start the process by filing FORM I-130 with USCIS. Once the relative petition is approved then the case will be forwarded to the NVC. Your spouse will have to file the FORM I-601A to request a waiver. Finally the case will be sent to the U.S. consulate for the immigrant visa processing. I recommend that you visit with an experienced immigration attorney to assist you.

Let's start with your status: USC? LPR? Something else? One thing I can tell you ... he'll have to leave the US to get the proper papers. Talk to an attorney in private via Skype.

He is probably not eligible for a green card but he may be eligible for a provisional waiver of unlawful presence and later for an immigrant visa to enter the US as a permanent resident. Work with an immigration attorney to understand the process and file appropriate documents step by step.