Food stamps and Medicaid are administered by the individual states. I don't believe you can use your Illinois card in other states because each state has a certain amount of money in their bank account, and when you use your card, you draw from that account. You can, however, apply in another state if you become a resident of that state. You have to have a physical address in a state to be a resident.
It is possible to use the Illinois link card in other states. You get to access it in the same way you do it while in Illinois. This is because it is of the same program in all 50 states but administered differently.