This is such a difficult situation to be in as a spouse. I am not going to weigh in about the possibility of depression, other than to say that it needs to be diagnosed by a professional.
Years ago my husband was in this same position (although the rehire didn't seem to be as possible). I went through many of the same things you have - follow up on when he will start looking, questioning why he wasn't looking, telling him to start looking & applying.
But what happens internally with our husbands when we act this way is detrimental to what we would like to achieve. Men like to be able to provide for their families, & when we question everything they are doing (or not doing) it chips away at their self-esteem. The natural response of anyone when that happens is defensiveness.
My suggestion to you is to take stock of where you are financially. What are your bills each month (mortgage, utilities, car payments/credit cards, groceries, etc.) as well as intermittent bills (insurance premiums, vehicle registrations, etc.)
Put this down on paper (I like to use Excel to organize & sum). Then, look at your savings. Your husband has been getting unemployment all this time, so your saving shouldn't be completely depleted. But you need to know where you stand.
Then, calculate out to the point when his unemployment will run out, & see just how long you can make it without income. This is your make it/break it point. And the unfortunate thing is, at this point, you have NO MONEY. So there is an urgency for SOME money to come in the door before this point in time.
Once you have all of this information, you need to sit down with your husband. Show him what you have put together, & walk through it. What it means that he is drawing unemployment now instead of working. What it will mean if he is not employed with a paycheck before the unemployment runs out.
Then, turn the question to him "what do you think we should do?"
And let him answer. If he says "I should have heard back from my company by then" ask him what happens if he doesn't. What is his "Plan B". ASK him, don't jump in & point out the problem you see with that approach.
The goal is to be collaborative in your approach. Let him know you are supportive of him wanting to pursue a course that leads to his ideal, but communicate that you need to feel secure that if things happen beyond his control, he has a plan & is willing to put it into action to make sure his family is secure.
One thing to make sure to discuss is the point where he will begin applying his plan B - looking for employment elsewhere. Because waiting until you have no money is not an option - there is a hiring process, from applying, to interviewing, to hiring, to first paycheck. Reasonably, it is about 1-2 months.
Make sure you are prepared to talk about your contributions - what will change financially if you take on a job. And don't just think about the money coming in, because if you are unable to make enough money to fully support your family, he WILL need to get employment as well, and there will be cost incurred for childcare. Other options - cutting out extras (cable, dining out, movies, etc.) and scaling back on necessities (if he is not working out of state, maybe 1 vehicle is sold, cell phones reduced to cheaper plans, etc.).
So, your purpose is to work with your husband, show your support of him, while also asking him to make sure you feel secure & cared for. Love is about give & take - sometimes we need to give of ourselves first before we can take what we need when it is offered.
Best of luck. T.