nope, veggies are not nessicary for a suplimental/treat feeding.
dogs are carnivores by classification and cant digest "normal" plant matter...
their can make use of greens (ie veggies) if theyve been cooked, or "partially pre-digested" (ie in the form of green tripe or in the form of all wizzed together in a blender...)
but because thier bodies arnt evolved to digest raw veggies i personally dont feel they truly NEED them if being fed an otherwise balanced diet.
My dogs however do like carrots and peas and green beans and a bag of frozen mixed veg is under $1 at walmart and bulks it out a little

that being said the gains (rice) arnt nessicary either but again, it bulks it out.
as a treat/supliment i feel you can mix whatever you want (asuming dog safe) into it, making sure its mostly meat of some kind...
if i were feeding this as a main food source id be putting alot more thought into balance.
as a main meal it would need more meat, organs and bone, less grain and veggies would be deterimend by if i was willing to feed green tripe or not. they do nd some vegatable matter as part of thie rmain diet, but it has to be in a "natural form" so green tripe is the best way, if thats not doable cooked or "veggie mush", or simply a good multi vit...
but again thats only if this was to be a "main diet"
i add the garlic simply because it supposedly helps keps fleas at bay and in small amounts it has lots of health benefits ust like in people.
generally when i make mine i add garlic powder to the water im cooking my chicken in, boil the diced/ground meat and then i use the water/broth from that to cook the rice in, so the rice ends up taking in most of the garlic/chicken nutrients.
once the rice is cooked i fluff then add a good drizzle of good olive oil and mix it well, helps keep the rice from stickign and makes a heck of adifference to the coat

flax would work too...
they love it
the general rule of thumb for dogs is they are carnivoers (some will tell you there not because they do eat veggies, but the fact of the matter is dogs ARE carnivores but unlike cats they are not obligate carnvores, meansing they can digest some plant matter under the right circumstances. in nature dogs would only eat grass/grain/vegatables that are left in the intestine of the prey animal...
dogs in the wild however will occasionally graze on ripe berries, fruits ect. this doesnt make them omnivores because they couldnt survive soley on those items...but they can and do enjoy them.
if you were feeding 100% home made food youll get lots op opinions on ratios...most sites say 40% meat, 30% grain and 30% veg...
now for a suplimental feed id say this is fine...and for a "general feed" it would be ok too...(its pretty much the ratio most good dog foods stick to)
however ideally a true balanced diet it would be more like 90% "meat" and 10% veggies...and of that 90% meat it should be about 10% organ meat (kidney, liver ect) 10% bone and 80% muscle meat and the veggies would be "pre-digested" (so partially cooked, ground/mushed or green tripe)
raw is even better, but i dont suggest mixing raw meat with kibble/cooked meals, they digest at different rates...
right now my mix is Meat/grain/veg about 40/30/30. and im hoping to bump it to 50-60/20/30-20