Since, the company included the reimbursement as income on your 1099-Misc, then clearly this computer is your property! Effectively, it would seem that you atcually paid for the computer. Thus, after you are done working for your client, the computer is yours.
Also, you were correct in deducting the computer on your tax return. I am hoping you claimed the section 179 deduction for the acquistion cost of the computer and deducted this entire amount on your tax return.