The harder I push him to find a job, the less time and effort he will put into looking, as if to prove that he won't be hounded into anything. He responds to my anxiety by shutting down--sleeping more and more, doing less around the house, and certainly doing less job hunting

Sounds like anxiety/depression on his part.
Would job-hunting to-do lists help for either of you? (For you to define what things are in your domain (and out of it), and for him to not be overwhelmed?)