There are a few ways that genetics interacts with diet. The study of these interactions is called nutritional genomics. Nutritional genomics has two sub-disciplines, nutrigenomics (the study of diet's effect on gene expression) and nutrigenetics (the study of how genetics effect response to nutrients). A second relevant area of study is on genetics and perception, especially relating to smell and taste. I'll give examples of two studies which found alleles (genetic variations) that relate to a veg*n diet.
Genetic variation for efficient omega-3 and omega-6 processing
Positive Selection on a Regulatory Insertion–Deletion
Polymorphism in FADS2 Influences Apparent Endogenous
Synthesis of Arachidonic Acid
This study found an allele which evolved in populations which have traditionally eaten plant-based diets. The allele's presence is associated with vegetarian diets and its deletion is associated with marine diets. When present, the body is able to more efficiently process omega-3 and omega-6. The high omega-3 levels in marine based diets make this allele detrimental.
A consequence of this and other genetic variations is that some people may have difficulty obtaining healthy levels of some nutrients on a veg*n diet and may need to suplement specific nutrients.
Genetic variation can change scent perception of meat
Genetic Variation of an Odorant Receptor OR7D4 and Sensory Perception of Cooked Meat Containing Androstenone
This study found that genetic variation affects the ability to percieve androstenone, a chemical found in the meat of male pigs, which is responsible for boar taint. Individuals who are more sensitive to androstenone rate the cooked boar meat less favourably than those who are less sensitive to it.
Genetic variations such as this one can affect a person's eating preferences, making them more or less likely to eat meat. (Further reading)