Can you apply for citizen ship with an expired green card?

yes in 10 years

No,you need to file Form I-90 first for green card renewal else you would be considered 'out of status' though permanent residency is valid and permanent. To apply for US citizenship your green card should have at least 6 months of validity left. This is also important if you plan to travel abroad or in case you need to prove identity for any purpose. Its important to carry a valid green card always else you will be fined or even jailed.