Employment law questions? Ask an employment lawyer.
There is no law that they have to give you a written warning. As an at will employee (which you are unless you have a contract) you can be fired for no reason or any reason at any time. The exception is that you cannot be fired on the basis of an illegal reason. Illegal reasons are discrimination based on race, national origin, color, gender, religion, age or disability- or in retaliation for a legally protected act, like whistle blowing or taking FMLA.
So yes, they can fire you just because they don't like you.
But, you should be able to get unemployment since you were not let go "for cause." Unemployment is available to anyone is unemployed through no fault of his own. Generally if you quit or are fired for violating company rules, you can't get it. But you should be eligible.