Compete Agreement after 3 years

I live and work in the state of Florida. I have been with my current employer for 3 years. I was recently offered a job working for a client of my current employer. My boss got wind of this and demands I sign a non-compete agreement and if I refuse "You will be terminated".
The new job is a supervisory position, watching over contractors that are awarded jobs witin the community that has offered me this new job. My current employer is one of the contractors that is occasionally awarded jobs in this community due to my representing them.
I have been offered a position monitoring and being a part of the choosing of future contractors.(different job responsibilities)
My questions are:
If I refuse to sign, can I be fired legitimately?
If I am fired due to not signing, will unemployment be available to this specific situation until the new job becomes available?