No, you can't use 首 to refer to students who have just graduated and don't have a job yet.
Pardon me if I am incorrect here, but I think maybe you think that "首にする" means "turn (someone) into a 首", and so 首 is an idiom for "unemployed person" or something. [Edit: My apologies, it looks like you were just taking the word of EDICT, which (wrongly, I think) makes exactly this claim.] This is not the case, though. "首にする" ultimately comes from the metaphor of beheading (!) and although it can be extended to other forms (もう首だろう etc.), it only refers to the act of firing/being fired, not to the result (being unemployed). So, it is not relevant to recent graduates... or any unemployed person, actually, except insofar as you can talk about their past: "She was fired in May," etc.)