No. The Idaho State PERSI is the state's retirement system; this is not the same as the federal government's Pension Benefit Guarantee Corporation.
Most taxpayers do not qualify for the Health Coverage Tax Credit (it's for workers who have been certified as "displaced workers" - see this TurboTax FAQ for more information.
If you don't know what this credit is or what this program is, then you are probably not eligible to take this credit.