It seems to me that Tom is misremembering the scene as it appears in the movie. Ennis couldn't possibly be secretly planning to meet up with Jack, because he and Jack haven't reunited yet.
However, I think I remember a discussion of this scene, either here or at IMDb, where we either had reason to believe or were just speculating that this scene was initially intended to appear later in the movie, where it COULD be about Ennis lying to get together with Jack. Even then it would be a little weird, if he's expecting to be home for dinner.
If that's what the scene was originally about, it's strange that they included it after all but in a different place. It doesn't seem very pivotal, and it's always made me a little uncomfortable because of the way Ennis mispronounces "girls."
It's interesting to see the range of reactions here, from feeling that it shows Alma being publicly humiliated, to feeling that it shows Ennis being in a tormenting situation. I just see it as the sort of conflict that most married couples get into from time to time.
Yes, Ennis feels that Alma's job is less important than his. Or at least, it's more urgent that he go to it immediately. And he very well could be right, judging from their responses -- Ennis seems stressed about getting there, and he obviously can't take the girls with him, whereas Alma has a sister and an accommodating boss and is able to take the kids. More may be at stake with Ennis' job -- it may pay better, and be more essential to supporting the family.
The other important aspect of this scene is Alma Jr.'s line, "I need crayons." As usual, so much is contained in a seemingly throwaway line! It's a metaphor for the whole movie.