Can I become a US citizen if my husband owes money back to unemployment?
You should answer all questions on the application truthfully and completely.  If your husband and you are filing taxes together, he should probably take care of any money he owes to IRS before you apply for naturalization or at least make sure that he works out some payment plan to take care of this so that you can present evidence of this plan if and when you are asked about this by USCIS.