X resigns from his company; Within a week he wants to get back to his company again. Is it acceptable socially

Answer 1

Yes it is. But other employees generally hate it, because it demonstrates that opportunities on the "outside" are lacking, and therefore it lessens their abilities to negotiate better benefits or salaries with their employer. It is indicative of an "employers" market when an employee can't find anything better on the outside.

Answer 2

Heck no.

X ought to make an appointment with a therapist and find out why he made such a destructive and impulsive decision in the first place.

What employer would trust X to stick around the next time he gets a wild hair? Not me.