Can my employer count Christmas Day, Boxing Day or New Year's Day as part of my paid leave entitlement?

In many cases, they can.

Some employers add the 8 bank and public holidays, such as Christmas Day, onto the minimum leave entitlement (5.6 weeks per year, or 28 days for those working 5 days a week). You should consult your contract of employment to establish the exact position in your company.

On the other hand, some employers may actually include bank holidays and public holidays in the minimum entitlement. This Scrooge-like approach may be mean, but it's not illegal - unless your contract of employment says otherwise, or you are being treated differently for an unlawful reason (eg because you work part-time).