Florida law states that employees may be eligible to receive unemployment compensation benefits if they:
1. Are totally or partially unemployed through no fault of their own
2. Are registered for work and file a claim.
3. Have sufficient employment and wages (usually )
4. Are ABLE to work and AVAILABLE for work
So, the answer is YES, but you can't quit your part time job and file for unemployment. You would have to file while working there which is not the best situation. You won't be very popular with the boss. You could be looking for another job while working part time and hold off applying for unemployment unless it looks like you can't find any other full time job within your skills and interests. If you file for unemployment while working and the boss fires you, you will have 2 claims: the first is for unemployment and the second is for wrongful discharge. See a lawyer who specializes in wrongful discharge under Labor Lawyers in the yellow pages or call the Florida Bar or your local bar association for a referral.