The solution that I came up with for that teacher was to not expect the student to finish the whole assignment before they get to their reward. Now you may say that is not fair to the other students, but life isn't fair. In education, you have to do what works best for each student. If you can get that student to finish 1/4 of the assignment and then let them get that reward, they will see that doing their work actually does pay off. Continue having the student complete 1/4 of their work before they get the reward for a week or so. After that, start increasing the amount of work the student is expected to do before they can get to that reward. After a while they will be completing all of their work and still having time for that reward. But if that student never gets to experience the reward in the first place, they will tell them self that they don't really want the reward anyway.
A good way to get students to buy into your reward system is to let all of the students experience the rewards to start with. If the reward is to get on the computer when they finish their work, let all of the students get on the computer at the end of a lesson. When all of the students see that they enjoy having that free time, or even the chance to play an educational game, it will make them want to play it more. After they have had the chance to experience it, then you need to start enforcing the rule of finishing their work correctly. This will help to eliminate the fight with those students who say they do not want to participate in the particular reward.
If a student truly does not want to use the computer, you may need to find a different reward that is more appealing to that student. Other ideas could be to give them extra recess or have them do an activity with another class. You have to get to know that student to find out what it is that motivates them and use that as their reward as long as it is something within reason. The whole idea behind a reward system is to motivate the students to do what is asked of them. If they do not care about the reward, then the system is not going to work properly.