How do I do this?
I want to share a page with a client. And then I’d like to add a login form to a website for the client to login and go to this page. I know this can be done with Basecamp. Can this also be done with Backpack? How?

That’s outside the scope of what Backpack. You could share it with them by sharing a page and inviting them by email, but they will create their own BP account. The login url will be dependent on what their account is, so you couldn’t link to it from a page.