No, usually if YOU quit a job, you cannot collect unemployment.
Next, does your Hubby even "want" to find another job? He does not seem to want to. He seems to like his job and his skill set and is specialized in that. So, what will talking to his Boss do? Finding him another job within the same company? And, the job has certain demands to it... and his living away from home. So well, that is the way it is. The Bosses cannot make special arrangements for him.... and have him be at work less and at home more. What about the other specialists like him, at his company? What do they do?
Also, perhaps your Husband likes his job and the travel/location of his job and being away from home. Did you consider that? Maybe he likes not being home...and having 2 lives. Because... if he did NOT like it... HE would have found something else by now, ON his own.
This situation of "you" trying to find another job for him... seems to be your idea? And not his. There is the futility in it all.
I know... him being away is not good for you or the family nor for a marriage. But if he does not come to that same feeling or conclusion... then I don't see how it will change, on his part.
Any person... if THEY are REALLY hating their job or unhappy... WILL do something about it and find other jobs or alternatives. Your Husband.. is not displaying those wants. Himself.
He seems to like his existence.
That is why he gets irked and chews you out when you suggest any possible leads.
Also... be very careful about "your" finding a job for him... because Employers, do not like it when a spouse is interfering in a job process or hiring... it does not create a good impression, for the one looking for a job. Head-Hunters, do not look favorably on it.
All the best,
Susan