Severance and unemployment regulations are one of those issues where the individual states vary wildly. So no one here can make a blanket statement about how it will be treated without being "wrong sometimes" Your state's unemployment workers will know exactly how to treat it.
Regardless of whether you feel you were discriminated against because of being on FMLA, in other words if you were the only one laid off, and you were laid off pretty obviously because you were on the FMLA, or whether your company is laying off lots of people and you are just one of the many, it's still nutso to refuse a severance package if you are entitled to one.
The company cannot keep you from getting unemployment benefits, it is going to be decided by decision by the unemployment system.
As far as their laying you off while on FMLA, you would address this problem nowhere with the unemployment system anyway, they totally separate issues. And you do need to get the unemployment wagon rolling as soon as you are laid off, legally or not.
As swalsh said, file the claim for unemployment immediately. DO NOT sit around and wait and try to "save it up" because you got the severance or don't think you'd qualify right now. Because the most important thing for you to do is set up the claim. Inform them(unemployment system) exactly how the severance is going to be, what you are getting and how it is to be paid. They'll make a decision on it. Then they will go into your medical issues, your health situation and whether you are able, available and fully ready to take another full time job right now and to make a job search for this new job while receiving unemployment benefits. If you're still compromised, you'll probably have to wait until you are released to start drawing weekly benefits.
While you may be able to receive severance pay while on medical leave and under a doctor's care, you will not be able to draw unemployment benefits. But your unemployment benefits will start when the severance is dealt with as it is supposed to be in your particular state, (in some state it is totally ignored, others are different) and when you are fully released by your medical doctor.
This gap between when you file for unemployment and when you begin to draw it does not mean you are losing weeks of benefits. Once filed, a claim is in place for one full year. You cannot save it up by not filing for it. They will never go back and pay you from the time you were off if you have not filed a claim. But once you have filed and your claim has been approved, those weeks are sitting there waiting for you to meet the the criteria to draw them.
If you think there is some issue where the company has singled you out to lay you off because you were on FMLA, you would need to talk to the EEOC about this issue, completely separate from unemployment.