Probitonary period

No company including any hospital should ask resign during this lockdown period. And, looking at your husband case, it's Head of the Dept (HOD) who is creating this issue, though I don't know the other side of the story. Your husband should speak HOD and also HR official or coordinator transferring him to other dept if possible. Secondly, there are many other hospitals and clinics in Singapore where are many job openings are available. He may opt to this if there is no way he can work in his present work place. As you didn't tell his profession, but any medical related jobs are in demand at this time.

I'm surprised that all his colleagues too behave differently. Good luck

In Singapore it makes little difference whether you are in probation or not, as the legal protection against being laid off is weak. Still, there is no "hire and fire" mentality, as e.g. in the USA.
Your husband should NOT resign himself, unless he wants to leave the job (and optimally already has another option lined up).
But in any case, he should have a Plan B ready for the case when he is laid off (potentially with little notice). This can happen to anyone at any time and it is best to be prepared.
Other than that, it is difficult to advice you. Sorry!

This all started when some local singaporean working in the same hospital wanted to join the department..the hod who hired my husband has joined some other hospital due to covid-19 situation and the new hod is not very fond of him and we have no clue why is that...my husband already spoke to the hod and she said that it's like this only in this dpt and if you don't like it you are free to go..the same things his colleague told him to do(i.e to leave)..that better you leave as you are not able to adjust here..and my husband told them that he is not going anywhere..so with this they are making his life more difficult by giving him extra duties and making sure he hardly gets any leave. We do understand that dere os some problem with the dpt as the last doctor left within 8 months of the joining but we are ot in situation to go back to India and start again as my husband left his prestigious job at aiims delhi. They have been denying my dp visa from ages and trying best to make sure my husband leaves his job coz to fire him they need solid evidence that he is not good in his work.
We have not yet explored if he can join any other hospital and will start exploring it from this week.

My husband is a pediatrician.

He is trying his best not to resign himself even though his life is very difficult after joining the dpt. There are only 3 ppl in the dpt one hod and 2 resident physicians. Both Rp and hod wants him to leave and they are forcing him to do it on a regular basis. My husband can't change the job due to new regulations by singapore medical council. This is the toughest phase in our life as I left my job and still on STVP in Singapore and we sold each and everything before coming to SG. All they want is to leave the job so that their friend can join..and they are not able to understand how bad it is going to affect us.. I wish people utilize the power they have in a proper way rather than destroying someone's life.