Can I be fired if I refuse to sign a non-compete agreement and if so, would I be able to receive an unemployment ?

You may not have any rights in this case, other than the right to sign the noncompete and (hopefully) keep you job. Companies may require employees to sign non-competition agreeements, and they may terminate them for not signing; non-competition agreements have been consistently held to be valid things companies may require.

An overbroad agreement would probably not be enforced in the future. For example, restricting you from working for competitors in a 50 or so mile radius, or taking as private patients any one who is a client of your employer, should you leave, would be reasonable. Trying to bar you from working as an RN anywhere would be too broad and would not be enforced (though it might be cut back by a court to an enforceable level, then enforced). However, that's an issue for the future, if there is disagreement over the scope...for now, the company may indeed require you to sign, at the price of being terminated if you don't.