If my husband leaves the US with an expired visa, how can I sponsor him to bring him back?

This will depend on the length of his overstay and your individual circumstances. Without further detail I cannot provide an informed answer.

If he overstayed his visa by 180 days or more, he will need a waiver to come back.

Consider adjusting his status before he leaves.

Generally, there should not be any problem as students under F-1 visa do not accrue "unlawful presence" until USCIS or immigration judge say so. Make sure he does not have such "finding". If he did not apply for any immigration benefits or did not go to immigration court, he is fine. Therefore, even if he overstayed his student visa, he is not subject to the 3 or 10-year bar. You may file I-130 visa petition with the USCIS. Once approved, follow the instruction to complete the visa processing procedure and he will be asked to appear at US consulate for immigrant visa interview. Please note that this response to your question is very general. You must make sure there are no other issues, such as "student visa" abuser or unauthorized employment in the US, which will complicate the matter.

You husband may be eligible for adjustment of status. If he qualifies, then he would not be required to leave. If he is leaving for other reasons, then there are questions of whether he has accrued "unlawful presence". If he has, then his case processing will require a waiver, which adds to the overall processing time.

If his status was not terminated, then he has not accrued unlawful presence. This means no 3/10 year bar. So you follow the same path as anyone else sponsoring a spouse abroad. It normally takes about 9-12 months for full processing. Please schedule full consult with a competent immigration attorney.

The key is to finish his paperwork before he leaves the US. The entire process only takes about 3 months.

If you are a citizen, your husband can apply for permanent resident status without leaving the U.S. You should consult with an experienced immigration attorney before taking any steps.