First, if you make a good patty and refrigerate before cooking, there isn’t any need to use egg. However, if you really do wish to use a binder — and there’s no shame in it and can still result in fantastic burgers — then I suggest reading this guide to alternative binders from onceamonthmeals.com.
They list things such as faux egg made from flax seeds, but also chia seeds, an oil and water mix, or gelatin.