I'm a graduate student & don't work. I get student loans for tuition & living costs. My former home state claims I owe back taxes. Can they seize my loans?

No.  Student loans are not something that can be seized.  But, the state can attach your wages when you start working, bank accounts, or any other assets you have if you let them get delinquent enough.  It is highly advisable to contact the state and make payment arrangements. 

When you "go away to school", you are still a resident of your former state until you take concrete steps to change your domicile.  Being  away at school is only a temporary absence from you permanent home state.  This is particularly true, if your parents claimed you as a dependent. 

If you filed  tax returns, in the school location state (or any other state, e.g. summer jobs), and paid tax, your former home state will allow a credit, or partial credit, for the tax paid to those other states.  The solution to your problem may be to simply file back tax returns for your former home state.