Will he be held accountable who swore to boycott the computer for a week, then used it at night a day after the period ended?

The day begins at the break of dawn, and what precedes it is night. However, in matters of oaths, the intention of the swearer is what counts. So, if by "day" you intended what is commonly understood by many people (from midnight to the following midnight), then there is no breaking of your oath, as long as you abstained from what you swore to abstain from throughout the entire period you intended.