He tells her to shake hands with men and threatens to divorce her if she does not do it

Firstly: 

It is not permissible for a man to shake hands with a non-mahram woman, and it is not permissible for her to allow that. 

The fact that such hand-shaking is haraam is indicated by the report narrated by al-Bukhaari (4891) and Muslim (1866) from ‘Aa’ishah (may Allaah be pleased with her) who said: By Allaah, the hand of (the Prophet) (peace and blessings of Allaah be upon him) never touched the hand of a woman when accepting the oath of allegiance. When he accepted a woman’s oath of allegiance he said: “Go, for I have accepted your oath of allegiance.” 

This was the infallible one, the best of mankind, the leader of the sons of Adam (peace and blessings of Allaah be upon him); he never touched women although the basic principle concerning the oath of allegiance is that it should be done with a handshake. So how about other men? 

It was narrated that Umaymah, the daughter of Raqeeqah, said: The Messenger of Allaah (peace and blessings of Allaah be upon him) said: “I do not shake hands with women.” Narrated by al-Nasaa’i (4181) and Ibn Maajah (2874); classed as saheeh by al-Albaani in Saheeh al-Jaami’ (2513). It was narrated from Ma’qil ibn Yasaar (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah be upon him) said: “If one of you were to be stabbed in the head with an iron needle, that would be better for him than touching a woman who is not permissible for him.” Narrated by al-Tabaraani; classed as saheeh by al-Albaani in Saheeh al-Jaami’ (no. 5045). 

This clearly shows that it is haraam.  

Al-Kharashi said in Sharh Mukhatasar Khaleel (1/275): Shaking hands with a woman who is not a mahram is not permitted.  And Allaah knows best. End quote. 

In Haashiyat al-‘Adawi ‘ala Sharh al-Risaalah (2/474) it says: It is not permissible for a man to shake hands with a woman, even if she is very old. End quote. 

“Very old” means so old that she would not be regarded as desirable. 

In al-Mawsoo’ah al-Fiqhiyyah (29/296) it says: There is no difference of opinion among the fuqaha’ that it is not permissible to touch the face and hands of a non-mahram woman even if there is no risk of desire. This applies if she is a young, desirable woman. The Maalikis and Shaafa’is are of the view that it is haraam to touch a non-mahram women and they make no distinction between young and old women. End quote. 

Secondly:  

A wife is not obliged to obey her husband if he tells her to shake hands with non-mahram men, rather it is not permissible for her to do that, because there is no obedience to any created being if it involves disobedience towards the Creator. 

This husband has to fear Allaah and beware of His wrath and painful punishment. This command of his goes against the command of Allaah, and his threatening his wife with divorce is unjust treatment of her. He should be happy that she is obeying her Lord and he should help her to do that.  

Thirdly: 

If the husband persists in his attitude, and threatens to divorce her, and insists on that, then it seems – and Allaah knows best – that the wife may shake hands with men from behind a barrier, and do the lesser of the two evils so as to ward off the greater of the two. But she should reduce the chances of meeting men as much as she can so that she can avoid shaking hands. She should also try to advise her husband and explain the truth to him, so that he will change his mind.  

If the husband is not convinced by that, and he insists that she shakes hands without a barrier, then she should not obey him, and she should be patient and seek reward with Allaah, even if she gets divorced, especially if this is a first divorce. Perhaps the husband will come to his senses if a divorce takes place, and will realize that he is destroying his family himself, for the sake of something haraam. 

She should also seek the help of righteous people among his family and hers, to resolve this dispute and convince the husband to change his mind, and remind him to fear Allaah and remember that He is always watching. 

The wife should also be certain that Allaah will grant her a way out, for He does not forsake those who obey Him and He defends the believers.  

We ask Allaah to set the affairs of the Muslims straight. 

And Allaah knows best.