(US examples, but likely applicable everywhere.)
The best type is one that pays in case you can’t do your job anymore, for any health reason. For them to verify what “your job is”, it’s easier if you’re employed full-time. Also, lack of steady income is more risk for insurance abuse, so premiums go up.
My disability insurance pays $10K/mo until 65 if I can’t program anymore, and there’s a 1 year period from disability to first payment. Premium is fixed at $2K/yr, and I started it at 35.
It’s a bit harder to abuse life insurance, but you’re still seen unfavorably if you don’t have evidence of steady income. Didn’t require employment verification, but the financial disclosures were extensive, and much simpler if you have a full-time job.
If you’re about to face a period of uncertainty, it might be better to face it with some liquid cash in the bank. It’s almost impossible to refinance unless you have a job, or 2 years of self-employment income. If you have home equity, consider a cash out.
If you think you’re going to live off your savings for a while, you may want to get them out of the stock market (or most of them). When I left my job, I gradually moved everything to US treasuries, but nowadays its all cash.
You also sleep better at night knowing that your critical runway is insulated from most of the randomness happening in the world.