They will certainly ask you about renewing your passport. You should have travelled on an asylee travel document or reentry permit. However it should not stop you from getting U.S. citizenship. They can consider your driving record regarding good moral character.
The above is intended only as general information, and does not constitute legal advice. You must speak with an attorney to discuss your individual case.
You should not have any problems with your citizenship application based on the travel.
The answer provided here is general in nature and does not take into account other factors that may need to be reviewed for a more precise answer. You should consult with an immigration attorney before taking any action. The answer here is not intended to create an attorney-client relationship.