If my husband had $0 income from self employment in 2019 due to illness, do I claim $0 bus. expenses? He should have income this year, so I don't want to say he quit.

If he had no Sch C income or expenses then you can file a zero income Sch C if you like. 

You mentioned that he had zero income and asked if you should claim zero expenses.  If he did have some business expenses that he incurred to keep the business open even though there was no income, you can still claim the expenses.  Otherwise those will be lost.  You would not be able to claim them on your 2020 return when he has income again.

@teray