Youre better off squatting somewhere than roughing it out in the cold in my opinion. Building natural shelter/camp or even a cave is also better than outside. if you really have to be out making a fire is almost mandatory, or heat packs, or something to warm you up.
Only way you can make outside liveable is to have high end mountaineering gear, which is expensive. And if your gear fails, gets damaged, wet, stolen, etc. your life is in danger
in levels of priority id say like this
squat > natural structure out of wood, or sticks/clay/mud/rocks > digging underground/into an overhang > finding a big log to fortify a camp under > cave...
as long as you shelter the wind and you insulate the ground its 100x better because you can put your tent inside it as well
if you have time now and its not too cold yet, bring an axe, cut some trees and make a hut, theres loads of tutorials online on how to make a simple structure without the need for nails and too much cutting
if youre having trouble finding a wild spot for a tiny hut, print out a page and say youre looking for a tent ground on someones property that you are prepared to pay rent (a small token amount) and/or send a bunch of emails to people asking the same
someone either gonna offer you a tool shed, barn, cellar, attic, their couch, the camp ground on their property or something else
just gotta make it believable and make people trust you, go all out mention references, who can vouch for you, and what steps youre making to get back on your feet and your plans, you would be surprised how nice people are when you ask for help and are genuine in your intent
think in terms if you had a property and why and how you would let someone make camp, it might not work for 100 people, but for the 101st it will, and thats all you need
but in my opinion, squatting for the win
once you get good at squatting the world is your oyster, just think how animals live, they roam around looking for adequate shelter and once not suitable anymore, they move on, like a cat just goes wherever, or a bird, adopting that mentality is liberating
just rambling some ideas and how I thought when I was about to become homeless
I chose squatting in the end cuz I didnt want to bother with annoying people who dont get the bum life and think its always something shady
so I squat in a greenhouse now (if its not an automatic/motorized one you can put the flaps down overnight and it gets nice and toasty, just leave a bit of an air channel gap) but yeah, for me it was either this, breaking into an unused building, or an abandoned building.
I just didnt tag them and survey for a few months before, so I had no idea how safe they are in the area
so I chose greenhouse, always open, no breaking in, and you dont damage anything, just put your tent down, sleep, pack up, and its like you were never there
edit: writing this up actually gave me the incentive to find some land to build on, the only thing I need is one of those foldable iron furnaces with a vent
that solves heating and cooking so youre not dependant on gas