Any diet can of course lead to deficiencies and hence health problems. If someone intends to make a significant change to their diet they would be best advised to ensure they will be getting all the nutrients they need (and nothing to excess). This is certainly possible with a vegetarian diet. Simply cutting out a significant element of one's diet (eg meat) without making suitable nutritional replacements is probably not a good idea.
That one goes both ways. Meat certainly doesn't contain all the necessary ingredients for health.