This is probably one of the most common questions I get. Unfortunately, I believe there is no "silver bullet" answer.
That being said, here are 6 common factors that come into play when looking at potential candidates. Each role will have a custom mix-and-match of these factors, but in general, aiming for a unique combination of them will likely benefit you in the job hunt, (at least for the first level of screening).
While many positions don't require a degree, it is definitely a preferred requirement for most UX jobs.
🙂 Good: Having post-secondary education
Having any kind of degree can be helpful, even if it is not directly related. It means you are capable of jumping the social and intellectual hoops of post-secondary education. Employers generally see this as some indication you have a "base-level" competency in writing, learning, and conveying ideas.
😄 Better: Having the "right" degree
While technically you can enter the UX industry from any background, entry level positions will generally favor people who have a formal education in something UX related. This could be HCI, Industrial design, systems design, etc..
There is no discounting the advantage in spending 4+ years concentrating your education into the field you want to work in, especially because you likely do not have years of relevant work experience to set you apart.
😍 Best: Going to a "prestigious" school
Like in all industries, there are a few really well-known schools that have a great track record of job placements. Some big names that come to mind are MIT, Carnegie Mellon, Stanford, UC Berkeley, etc.. While there may be a degree of correlation vs causation as well as self-selection bias, the reputation of a school or program is often something that impacts the on-paper attractiveness of a potential candidate.