I don't know about other states, but in the State of FL, you have to claim weeks that you were unemployed and you cannot refuse work. You also have to list jobs that you have applied to; whether or not there is any follow up to verify that someone actually was looking for work, is doubtful.
You cannot work and collect unemployment at the same time., unless you are only working part time and then you must report the hours and wages that you worked.
It is my understanding that an employer is supposed to report to 'unemployment' those people who are back to work and those who refused work.
It is understandable, given that people were given an "unemployment" extension which is $600.00 week for 4 months, in addition to their unemployment that they don't want to go back to work if they are making less working, however, right is right and wrong is wrong. Yes, collect the unemployment you are due for time missed because of your place of employment being shut down, but if they open back up, you best be ready and willing to work.