Just read, after writing this, that things seem to be getting better. That's great. This was what I'd written:
Loads of jobs are really stressful. It's likely that this is really another stressful job, and that he's not over-reacting. Unfortunately many companies are going that way - cutting costs and piling the pressure on their existing workforce.
First of all, try to be calm and don't panic as it may add to your husband's stress. He can still keep looking for jobs, to keep his options open. It is quite normal these days to change workplaces frequently, and it may not be viewed negatively if he changes jobs again. He could give a good, bland reason, like 'involved too much travelling/nights away/job didn't stretch me enough/job was not as advertised', etc. He could even miss this current job off his CV and pretend to his next employer that he just had a career break or a long holiday, living off savings for a while, if he doesn't want to include it (done that loads of times).
New jobs are ALWAYS stressful. He's probably been thrown into the deep end and it can take a year to get your head around a new job and feel comfortable with it. Reassure him that it was always going to be a difficult time.
He should speak to his manager (just read that he's done that!) to see if he can gain a clearer understanding of expectations. Also, he needs to set his manager's expectations - 'what are the priorities?' 'If I focus on X today, then Y won't get done until Friday'. Start sentences with "I need your help with...."; it makes it difficult for them to say no, and inflates their ego.
He should see a doctor and talk about how he feels. He needs to be really honest with the doctor. He may consider going on anti-depressants - they don't have to be a long-term solution (I've used them twice before for two particularly stressful periods of my life, for short periods), but they really do make you feel as if things 'don't matter'. It will have a calming affect on him to help him through this period (does not mean you have issues). They can have other side effects (for me though, the only negative affect they had was that it can numb you in physical ways - sex drive can be affected), so just bear that in mind.
If things don't get better, think about how you can relieve the financial pressure by returning to work / upping your hours / changing jobs. If you know you have options if the worst happens, it will help you feel more 'in control' and less stressed about the situation.
Hope that helps x