Don't know why you should provide screenprints to prove sign up, they can see that you've blocked them!
I cant see the harm in printing out the actually email from the job that you have just applied for. I have been doing this for the past year now. Its my way of providing proof that I am applying for jobs, and it keeps them off my back. I also save every email for my record.
I have learnt over the years, it is best to cover your ass.