Learn English – Try somebody out or just try them

Both you and your informant are correct in different circumstances.

The difference lies in the character of the verb involved. With a telic verb—one which includes an end, a specific goal or point of completion—there is a marked difference between the simple form and the progressive form: the progressive form marks the action as incomplete. Eat, fall, take are telic verbs.

With an atelic verb, which does not include an end, there is no practical difference. Stand, run, think are ordinarily atelic verbs (although in some circumstances they may be ‘recategorized’).

See the tag-wiki entry on aspect.

Lets note the tenses for to do, to finish and to eat (I added the last one because it's quite common and it's an irregular verb). So I'll list the infinitive, the past tense and the past participle:

To do, did, done

To finish, finished, finished

To eat, ate, eaten

Now, let's look at your first example:

I just done and I just finished

The first is obviously wrong, because done is a past participle which needs some other verb. This is not the case for finished, because it can be read as the past tense (rather than a perfect tense). Compare the following:

I just ate and I just eaten

Clearly, the first is grammatically correct, whereas the second is wrong.

The same argument can be made for your second example:

I ate and I eaten

The first is grammatically sound, the second is not. With to do and to finish this can be explained by noting that finished can be read as past tense (which is correct in this case), whereas done can only be read a past participle and the example lacks another verb to make it grammatical.

In your third example, I think both can be grammatical, but done is not used in the way you do. To finish dinner is idiomatic (people generally say that to indicate they have concluded their evening meal). To do dinner, however, is not idiomatic (it's not used by native speakers).