The skin should not break out after a facial, but there is a possibility that it might and is completely normal if it does.
There could be many factors influencing skin results after a facial, the process of extraction for example, where the skin is often exposed to bacteria from the sebum extracted, could cause a post-treatment breakout.
If an individual has a particularly sensitive skin, then there could be a reaction to specific treatments like steaming or stimulating massages. A reaction could also occur when new or different skin products are applied during the facial.
If in doubt, always mention concerns to your therapist, and insist on hypo-allergenic products for sensitive skin. If you do not know how you will react to a facial, then never book a treatment too soon before a special event or occasion.