My friend and I are writing a book in English in the Electrical Engineering field. At this stage, we need someone to proofread our texts and also to design the layout of the book, so we will need to entrust them with our manuscript. I was wondering, how can we trust that the proofreader and designer of the book will not plagiarize our work?
Work with reputable proofreaders and designers, who have proven track records with satisfied clients.
Anybody with a reputation to maintain will have no profit from plagiarizing clients' work. First of all, a manuscript on its own isn't worth a whole lot (it takes a lot of work to earn good money off a manuscript), and secondly, they'll stop getting clients.
In addition to "Work with reputable proofreaders and designers," as Standback correctly notes, you can also add a non-disclosure agreement to your contract. The language might state that the contents/cover/etc. of the book is to remain strictly confidential until official publication by X publishing house or Y printer.
Honestly, I don't think this is something you need to worry about. Any proofreader or designer who plagiarized clients would be out of business the next day. Nobody is going to work with someone who steals from the client. It's utterly self-defeating.
It is not clear from your question if you intend to publish your book with an established publisher or in an independent way.
In scientific fields, proofreading is typically done, at least partly, by colleagues who know the subject. Here confidentiality is commonly guaranteed by trust. Then there is the publisher's editor. Here confidentiality is guaranteed by the contract.
The book design should typically follow the publisher's style, which can depend on the book series. Many scientific publishers distribute style guides and LaTeX classes to their authors. If you want to follow your own design you should probably make an agreement with the publisher.
Finally, take into account that the yearly income which can result from selling a scientific book is typically very limited and hiring a professional proofreader and a designer can be onerous.