Startups may be less picky. If you want to work in startups it's probably more important to get involved in startup networks. I can't really say more than that as I don't work in startups.
For many other software jobs, a degree will be required. This is an applicant filter if nothing else. These places (banks, insurance companies, manufacturers, any number of business where software enables what they do but is not the product they sell), can be boring but they can be great places to work if you want boring. Regular hours, low stress, good benefits. They will almost all require degrees. From which institution might be less important as long as it's reasonably legitimate.
For many other software jobs, a degree will be required. This is an applicant filter if nothing else. These places (banks, insurance companies, manufacturers, any number of business where software enables what they do but is not the product they sell), can be boring but they can be great places to work if you want boring. Regular hours, low stress, good benefits. They will almost all require degrees. From which institution might be less important as long as it's reasonably legitimate.