Can I prevent students from clearing their browser history?

Yes. This can be configured in Google Admin.

Yes.  This must be done within Google Admin for the domain in which the student's user account exists - which is typically the school's domain.

Navigate to Devices > Chrome > Settings > Users & Browsers

Scroll down about 10% down the page, or search on the page for "Browser History".  Click the dropdown menu and select "Do not allow clearing history in settings menu".