This will be considered a debt owed the government. Those are generally not dischargeable (like taxes, not like student loans).
I also doubt that an undue hardship application would be successful.
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The debt may not yet be dischargeable because certain military enlistment bonuses if bankruptcy discharge is less than 5 years after the termination of an enlistment for which an enlistment bonus was paid if the person voluntarily or because of misconduct did not complete the term of enlistment for which the bonus was paid. Other military special pay and accession bonuses are not dischargeable (certain special pay and accession bonuses for pharmacy officers, certain retention bonuses for members of the Armed Forces qualified in critical military skill, and certain debts related to the Information Security Scholarship Program). PL 106-398, 2000 H.R. 4205.
Disclaimer: This answer does not constitute legal advice. I am admitted in the States of New York, New Jersey and Massachusetts only and make no attempt to opine on matters of law that are not relevant to those three States. This advice is based on general principles of law that may or may not relate to your specific situation. Facts and laws change and these possible changes will affect the advice provided here. Consult an attorney in your locale before you act on any of this advice. You should not rely on this advice alone and nothing in these communications creates an attorney client relationship.
You might want to do a Chapter 20. A Chapter 7 discharge which gets rid of all other dischargeable debt and then a Chapter 13, where you won't get a discharge, but at least the government is kept under control....and pay that off in 3-5 years. See an experienced consumer attorney up in Portland!