Nutrient deficiencies can happen whether you eat animals or not. ANY inappropriately planned diet can leave you with nutrient deficiencies OR nutrient excesses. And both can put you at risk of getting sick.
However, most people today are dying of nutrient excess NOT nutrient deficiencies. The leading cause of death around the world, heart disease, is caused by an EXCESS of dietary cholesterol and saturated fat. So don't you think that more people should be questioning nutrient excesses? No one really talks about that though, do they?
But I digress, the simple answer is eating a calorically sufficient quantity (enough food to maintain a healthy body weight) of a decent range of whole plant foods from all food groups (fruits, vegetables, nuts, grains, seeds and legumes/beans) - with a specific focus on vegetables, as they are the most nutrient dense food group on earth - will easily give you everything you need... with ONE exception...
The truth is there is no essential nutrient that naturally occurs in animal foods that can not also be found in plant foods from your local supermarket. In fact, just white potatoes contain everything we need! So this is not as big of a deal as people often imagine. Remember there are plenty of healthy, elderly vegans who have been thriving on a plant-based vegan diet for 50 years or more.
The sole exception to this easy to follow rule is vitamin B12. This essential nutrient is different from the rest because it is not found naturally occurring in any plant or any animal... it comes from bacteria that are found in the earth, in water and in the dirt. Eating foods fortified with B12 such as some soy milk's or taking an inexpensive supplement once a week will give you all the B12 you need.
What most people don't realise is that B12 deficiencies occur in about 38% of the general population... considering less than 5% of people are vegan, this is clearly NOT a vegan problem!
It is an issue of our clean, sanitized world that can easily be fixed for less than five dollars per YEAR with supplementation, and the rest of your nutrients can easily be obtained from a delicious range of easily accessible whole plant foods.
If you want certain proof for your own body though, probably the best way to judge whether you are getting all the nutrients you need is to get blood tests done and add plant foods or supplements in accordingly.
Any diet can be expensive if you eat at restaurants for every meal or only shop from overpriced "specialty" stores, but as a general rule, eating only plants is actually CHEAPER than eating animals.
The cheapest foods all year round are vegan foods. Rice, potatoes, beans, lentils and other vegetables and fruits can all be bought for only a couple of dollars per kilo in nearly all of the world. These foods are staples in some of the poorest places in the world because they are so cheap!
It's too much to go into here, but even if you break it down into cost per gram of protein, vegan foods are still less expensive than animal foods.
Calorie for calorie, if you want to save money, whole plant foods are the way to go. Charities know this, and this is why most organisations that feed the homeless feed them vegan food... because it's so cheap they can feed more people for less money than incorporating animal foods into their recipes.
So veganism is NOT expensive! This is a myth probably resulting from vegan mock meats and fake cheeses historically being very expensive, but as they become more and more mainstream this is quickly changing. And obviously, you don't need to eat these processed goods to be vegan...
(...I bet you weren't expecting that, were you?)
The truth is, veganism is NOT the most ethical way to live... not necessarily.
But it IS always a more ethical way to live - based on moral positions that most people, whether they are vegan or not, already hold, positions such as "it is wrong to harm animal unnecessarily" - as opposed to a meat-eating lifestyle.
To use an extreme example... there are meat eaters out there who are billionaires who donate millions of dollars to effective charities that save untold numbers of human and animal lives from suffering and early death. I would say these few people are undoubtedly "living more ethically" than nearly all vegans because the amount of suffering they are helping to cease with their generous donations is much greater than the average person who is vegan.
Unfortunately, not everyone is an altruistic millionaire... but everyone spends money every day, and choosing to spend that money in a way that is in alignment with your value of not causing unnecessary harm to animals IS clearly a more ethical decision than choosing to pay people to stab innocent animals in the throat everytime you get hungry.
So when it comes to an everyday choice that EVERYONE can make right now, going vegan IS the most powerful tool we all have in our grasp to make the greatest positive impact on the earth, the animals, and the environment, that takes up no extra time or effort than the things we currently do.
So in this one area, yes, veganism is a more ethical way to live based on positions most people already hold, but there are many factors to consider when making broad statements like "X is the most ethical way to live"... be it going vegan, donating to charity or anything else. Make sense?
Non-vegans have many questions for the vegans they know, and these questions vary dramatically depending on the person.
Unless you want to isolate yourself from all social contact for the rest of your life then just let meat-eaters know you're not some extreme weirdo who eats nothing but lettuce all day while conveying the most central message of veganism - animal rights - in an easy to digest way.