I'll do anything = I'm offering to do A, B, C or D. I'll do everything = I'm offering to do A, B, C and D. For example: "What would you like to drink? I've got tea, coffee, hot chocolate, fruit juice, lemonade and Coke". "I don't mind. I'll have anything. I'll have whatever is easiest for you!" By 'anything' you mean any one of those drinks. You are obviously only going to have ONE of the drinks on offer, but you don't mind which one. You would not say "I'll have everything", because nobody is going to ask for tea and coffee and hot chocolate etc all at the same time, are they? I hope that makes the difference clear.

There's relatively little difference between the two, though 'everything' might imply that you intend to do x action(s) alone, versus 'anything' implying that you are CAPABLE of doing of everything only (not necessarily that you want to do everything).