Can they ask me to reapply for a lower position than the one I actually had?

They can ask, but you're not required to simply accept a lower position just because it's what's being offered, and being asked to "reapply" for a demotion is functionally different from being kept on in your existing role. If your employment is treated as continuing through the sale and the new owner is genuinely only willing to offer you a meaningfully lower position — less responsibility, lower pay, reduced status — that offer of worse terms doesn't automatically bind you the way your existing job's terms did; declining it isn't the same as voluntarily quitting a job you were entitled to keep.

Depending on how significant the downgrade is, being offered only a materially worse position, rather than continuation of your actual job, can be treated as effectively ending your existing employment, bringing different entitlements into play than if you'd simply been asked to keep doing the same work. How you respond matters here — accepting the new, lower position without addressing the change can later be read as agreeing to it.

If you're asked to reapply for something clearly beneath your actual role, get the specifics of the offer in writing and get advice before responding, rather than reacting in the moment.