What's the difference?
Eating lunch and having lunch (or even eat food and have food)
There is a difference, but eat and have are often used to mean the same thing, and as a result native speakers (particularly with American English) do not really think about them too much.
If you have lunch, we are talking about the event, the institution, of sitting down at a table, being served with food and then consuming it. Using have instead of eat is also considered more courteous in conversation (and is often thought of being a British English phenomenon), particularly if you need to express yourself politely.
If you eat lunch, then this is just about consuming the food. Where you eat, what you eat with and what you eat on are not important.
In these contexts, you are more likely to have breakfast, lunch, and dinner if you are treating it like an occasion, such as sitting down to eat at the family table (with food being served up in dishes or bowls for everyone to help themselves), going to a restaurant, or simply being served food at a party, friend’s house, and so on: I’m having breakfast at the local café. I had apple pie yesterday at Anne’s. If you eat breakfast, lunch or dinner, you’ll probably have this around the TV, at work (probably with sandwiches), or even on the move (not so healthy) as the focus is having the food with you and eating it and not the etiquette as to how it is presented: Let’s eat lunch at Benny’s. Do I have to eat brussel sprouts? Brian, eat your vegetables! In addition, you are more likely to eat rather than have food at a fast food restaurant or in a canteen, as there is usually no formality or ‘frills’ as to how the food is being served and are considered unimportant or irrelevant.
The meaning, however, is more clear if, for example, you offered somebody a sweet (or crisps or any other snacks in a bag). You are more likely to say ‘have a sweet’ or ‘would you like a sweet’. If you say, ‘eat a sweet’, this is an instruction; ‘would you like to eat a sweet’ does not always mean your sweets, and would sound strange to the listener. But it is possible that the speaker could ask about how the listener feels at the moment: If you’re not feeling too well, eat one of these fruit sweets.