It's not okay for your boss to try to "fix you" and force you to implement changes in your life that you are not interested in and do not need.
If you have a health problem, it is up to you to solve it. Your boss's involvement should be limited to expressing concern for your well-being, and following the law and company policy about paid leave. His belief that lack of breakfast is making you sick is not well-supported by science.
Employers have no right to dictate your behavior outside of work. While they can make requirements that might surprise you (like only hiring non-smokers, because smoking is not a protected class), I doubt they can make a requirement for you to eat breakfast, if only because they have no way to know whether you have done so, and furthermore employees would rise up in arms against them for such intrusive and controlling policies.
It may be fine for your boss to share his concerns once and suggest a possible solution, whether he's factually right or wrong. This can be a simple expression of desire to help. But going past mild influence into manipulation or outright coercion is unacceptable.
I recommend you stop giving your boss information about your eating habits. Start by no longer volunteering any information at all. If pressed, answer that you've eaten everything you need to eat up to that point in the day. You're not hungry. You're feeding yourself properly and you don't feel that your performance at work should be judged on your personal eating habits. Keep answering the same way over and over.
If your boss were demanding you go to the restroom and eliminate at certain times or with a certain frequency, would that be okay? While companies can make policies about when breaks may occur based on the normal capacity of healthy adults to wait between bathroom breaks, would you tolerate him requiring you to inform him of the exact nature of your bathroom activities and quantities produced? Must you go to the bathroom on your break and "try" if you don't need to? Maybe you can see what I mean. This is invasive.
If you are an adult, he should treat you like an adult—and you should act in a way that makes him understand that he can't treat you like anything other than that. You do not need to explain your eating habits to him.
Abide by your company’s policy, follow the law, and do what is good for your body and makes you feel good physically. Your employer should butt out.
I'm going to add some info about the health science, and am also adding some advice that is only here because it seems to me that the best solution for you is to be sick less often. You should of course do your own research as I have no authority—these are just one-time suggestions for your review.
I'm unaware of any direct and objective evidence that adopting the routine of having breakfast (the first meal of a person's day) around noon has any guaranteed deleterious effect. It is true that some people suffer if they go many waking hours without eating, but it seems clear that you do not.
It is likely a problem to skip breakfast if you normally eat it, but if not and you feel good, then there is no issue to continue doing so. You are essentially having your breakfast at lunch time, and there is nothing wrong with that. Many societies in history ate only a single meal each day and they did fine. Some societies today still do it!
Are you getting enough vitamin D? The next time you have a blood test, just ask for vitamin D levels to be checked. If you are low, then this is the easiest thing you can do to materially reduce the number of times you get sick. No, it is not enough to go out in the sun unless you are getting a full hour of sunlight with the sun directly overhead (which is not possible past 23°26ʹ degrees N/S latitude). Also, note that vitamin D is really more of a hormone than a vitamin, and if you do try to get it through sun, know that it is made directly from cholesterol, so if you are eating in a way that you have low cholesterol, this could impact vitamin D production. (And for that matter, LDL functions as an immune defense mechanism so very low levels could be an illness risk.)
In support for vitamin D supplementation claims: Harvard.edu Supplements Scorecard
It's very hard to get the vitamin D you need from your diet; oily fish and fortified dairy products are the only important sources. So supplements do make good sense for most adults. The form known as vitamin D3 is usually recommended, but D2 is also effective; for best results, take your vitamin D along with a meal that has some fat. If you want to be sure you need this supplement, ask for a blood test; levels of at least 30 nanograms per milliliter are considered best.
Or better, The BMJ – Vitamin D supplementation to prevent acute respiratory tract infections: systematic review and meta-analysis of individual participant data
Vitamin D supplementation reduced the risk of acute respiratory tract infection among all participants ... protective effects were seen in those receiving daily or weekly vitamin D ... Vitamin D supplementation was safe and it protected against acute respiratory tract infection overall. Patients who were very vitamin D deficient and those not receiving bolus doses experienced the most benefit.