Is it right to use a singular form of the verb in this sentence?

This is one of those situations where there is only a less bad solution. The less bad solution here, in my opinion, is to follow the proximity principle and write:

Either the teacher or the students were enjoying the picnic.

In your position, however, I would advise your students to avoid the construction altogether. There is a good section on this and similar problems of agreement in The Right Word At The Right Time (p31):

• ?Either the birds or the postman wake me in the morning.

• ?Neither you nor she ever wake before 11.

In both examples here, wakes would be preferable. When there is a compound subject like this, the verb takes its form from the noun that is nearer to it. Had the elements in each subject been reversed, then wake would have been correct.

Many people feel that this is still not a satisfactory solution. The impasse remains. The best course then is this: since it is the structure of the sentence that causes the impasse, simply change the structure of the sentence:

• Either the birds wake me or the postman does.

Keep trying new constructions until you are happy with the wording. English is a remarkably versatile language. One version after another might be unsatisfactory, but persistence will turn up a suitable construction in the end.

The obvious restructuring in this case results in:

Either the teacher was enjoying the picnic or the students were.