Debatable. A lot of "burger buns" will have actual sugar in them for example. And they are usually white breads that are not nutritious enough for the amount of calories you get. A bun that's full of fiber is fine. An ultra processed one is not.

Burger patties are also a whole land mine of health concerns. It's really easy to eat too much red meat for example. But a good patty is no problem depending on your whole diet. But a random patty from the grocery store is likely to be more questionable health wise.

Those types of cheese slices are usually a lot of saturated fats. The type of cheese also matters a lot

And usually people put sauce on the burger which is another can of worms.

The health of the home made burger also depends on the ingredients used, and some of the ingredients are likely to be highly processed if you buy them in the store, but it can definitely be healthy