eslHQ - Future tenses

First of all, these aren't tenses - tense involves morphological marking of the verb - eg addition of 3rd person -s of past -ed - and English only has two - past and present. So both of these verb forms actually have [i]present[i] tense - and, in fact all future events in English are expressed from the point of view of present time. We could paraphrase your examples as:

She intends to tell Jack that he...

I predict a bad grade for my test.

In other words the speaker is expressing a present intention or making a prediction at that moment - the present. It's only the event itself which is future, and that's expressed by the infinitive in both forms

Both verb forms will + infinitive and Be going to + infinitive can express either intention (including willingness, insistence and other forms of volition) or prediction. So yes - either form would be possible. It depends on the context. The only thing I'd change in your examples is that She will would almost always be pronounced She'll unless there was strong emphasis on the modal verb - eg contrastive stress - and would very often be combined with expression indicating the speaker's degree of certainty : I'm sure she'll tell Jack that.... / I think she'll probably tell Jack that...

Obviously it's a bit more complex than that sometimes - you happen to have picked examples which are interchangeable. But it would take too much space and time to explain here and (by pure chance!) I've written two articles about it on the ELT Notebook - so I hope Eric and Little Sage won't mind if I link to them:

Understanding Will
Will and Be Going to : What's the Difference?

Have a look at them and if anything is still not clear, get back to me here.