No they can't. They owe me bitcoins, not dollars.
After being screwed by several exchanges over the years, I can sympathize. But under the law, that's not true. Fiat currency generally must be accepted as payment for debts. Technically, that's what fiat currency means: currency "by decree." Not being familiar with the Japanese language, I can't point to the exact law, but it has parallels with all major fiat currencies.
This is what the US Treasury says about credit obligations vis-a-vis the dollar:
This statute means that all United States money as identified above are a valid and legal offer of payment for debts when tendered to a creditor.
As a creditor, you have no choice but to accept fiat currency. This is why it's so incredibly important to control your own keys. If Coinbase gets hacked, those coins are gone forever. You might get some $$ value back in bankruptcy/reorganization, but those bitcoins can't be replaced.
