Disclaimer: This is coming from an American Bird learning to cook from a bachelor British bloke and his mum., both of whom are good cooks.
Bubble & Squeak is just leftovers mushed together and then heated up in a pan. Like @Bellatrix says, there are no rules for the ingredients, although potatoes and cabbage are traditional. It’s all about maximizing the leftovers and turning it into another meal.
Feel free to add anything you want. Chopped fresh onions are fine and would add to the basically bland taste. Once mixed, form it into patties. I find it helpful to stick them in the microwave to warm them up before frying. They should be hot, and the frying doesn’t take that long. It doesn’t take that much butter or any other non-stick coating, like oil or animal fat. The goal is to get both sides slightly crispy and have the insides hot and still moist.
@Sunny2 Bubble & Squeak is fairly bland since all of the ingredients are just being rewarmed. If you are like me and prefer a bit of flavor added, just about any condiment can be used. It’s all about preferential taste.