I missed several days this year for family illness (my children and my husband had an accident) Also had jury duty and had 4 court days with the A.G.
FMLA does not apply. I have been with same company for 13 years.

My boss gave me the option of part-time work w/o benefits or to give my notice. Anyway I could qualify for unemployment if I quit to look for full-time work??

Almost certainly not. In the overwhelmingly large majority of cases, if you quit you don't get unemployment. In the rare exceptions, it is almost always the case that an employee is being subjected to illegal behavior of some kind, or (in just a few states) has left to follow a spouse to a new job beyond commuting limits. The state is not going to pay you to quit your job so that you have NO income instead of SOME income.

However, if you accept the part time position (while you look for full time work in the off hours) it is possible, depending on your state law, that you may be eligible for partial benefits to make up some of the difference.