ABSOLUTELY !! Any trading income shows up on your Schedule D. All Sched D income doesn't require SS Tax to be paid on it. To be totally on the up-and-up, just make sure that you don't claim any of your trading expenses as business expenses. Hence, you were just "trading" while you looked for another job.
I have to agree with the sentiment that the people at the unemployment office have no f**king clue what they're doing. Especially in NJ. But I think this carries over to most states. That said the racist part is a little over the top. There is no one group of people with a monopoly on stupidity. Other than ET posters that is If they somehow determine you can't collect it (if I remember correctly the worst they can do is ask for the money back) you need to demand a refund of all your unemployment payments to the state. They can't collect if you can't collect.