How do I avoid laying off any more of my employees?
Not knowing in what state you do business, I cannot say unequivocally that the following applies; but in most states, unless the individual is fired "for cause" s/he collects the same unemployment benefits as if s/he were laid off work; and your unemployment taxes will be affected the same as if you laid him off work. ("For cause" normally means egregious behavior such as stealing, threatening someone, etc., versus just poor performance).
So, whether you lay off more employees or fire them, they would be able to collect unemployment and for a good long time with the prevailing laws.
I would not be inclined to cooperate with his request, if I were you. I would tell him that I am doing the best to do the right thing by keeping people employed during the economic downturn and that you appreciate everyone understanding that fact and doing their best work until times get better for all.
Document his request to be laid off work (date, time, when, where, how it occurred) and your response--via email or otherwise in writing is best.
Then, keep track of his attendance and job performance. If he begins performing very poorly or not showing up for work or completing his shifts at work, you should document properly and, if it continues, give him an unpaid suspension documenting that you are giving him this time to draw his own conclusion about whether he still wants a job or not.
When he returns from the suspension, have him document his decision in this regard. You may have to do this more than once, but it lays the proper groundwork for claiming "self-termination" (or resignation) on his part which would make a decent case for him not to be classified as having been fired and immediately eligible for unemployment benefits.