Hi slothemma and welcome to the forums

Are you doing SW from home or are you going to group?
I think that because Slimming World is so versatile, you should be able to make it work for you, even if you are a fussy eater. They do recommend filling half your plate with S foods, which are fruits and/or vegetables but it's not mandatory, you're just likely to see better losses if you do.
My husband is doing SW with me and he's a fussy eater too. In the last 2 weeks he's started eating satsumas but before that he wouldn't even touch a piece of fruit. He's similar with vegetables - he'll only eat tomatoes (ok that's a fruit but hey ho) and broccoli really but he's lost 10.5lbs in 2 week without filling his plate with veg at every meal so it can be done.
Putting veg in soups/stews is better than nothing and all of that will count as speed food anyway. Do you eat vegetables in other dishes, such as chilli or Bolognese? They're good dishes to hide vegetables in, such as grated carrot or courgette as once it's cooked you can't see or taste it. They're also full of tomatoes, which are great speed foods. I make a lot of soup to take for lunch at work and it's all free on SW and really nice.
I don't think the egg thing will really affect you. They are free on SW and are filling so they're a popular choice but again, not mandatory. I hate hard boiled eggs and scrambled eggs so I don't eat them very often. There are tonnes of lovely recipes out there that are SW friendly without the eggs so I'd say give it a go and see if it works for you. I'm making the SW BBQ pulled pork in my slow cooker at home at the moment and that's free on SW but other than a little passata, no vegetables in sight

Pasta, rice and potatoes are all free too so plenty of options there without using eggs.