Of course. It's the very nature of life. Just because you don't have a degree doesn't mean you're locked in a certain career. It just means others will have a slight advantage changing their's.
It varies greatly based upon the field of work as well as your state laws. I think every state has a 3 month 'trial' period to where they can fire you without any reason. If it's within those 3 months then unemployment can't be collected. Usually there are steps that each company sets in place...
Go the degree route if you plan on pursuing any further eduction past your paramedic. If you don't plan on going anywhere, just get your certification.