Hi there. We use both and it means the same thing. The correct grammer is 「別の日」 which means a different day. I think for 「別な日」, the な is used like キレイな花、which follows an adjective verb (so in this case キレイな is an adjective verb) and it's a mix up. People will understand but saying 別の日 is better.

Since 別 is a noun, when used as a descriptor for another noun it must take の, and not な. So, only 別の日 is correct. For comparison, take the adjective 綺麗（きれい), which hence takes な and not の: 綺麗な日 and not 綺麗の日.