H2O2 is not for spraying on plants, it's a soil soak whereby when it breaks down into water and oxygen, it will form O2 molecules, unless there is fungus to attack, when it will oxidise and kill the fungus.
Fusarium and verticillium wilts are both soil-borne, so my hope, rather than expectation, would be that it might at least help in the short-run, and eradicate the problem in the long-run.
As for EB, I would imagine you'd have to soak your soil periodically, but it would help against infection by soil-splash, even if it wouldn't help with the wind-borne element.
For that I'd use foliar a-a-c-t. with double molasses.