Can purification columns from other suppliers be processed on the QIAcube Connect?

No. The QIAcube Connect is designed for fully automated processing of up to 12 samples using QIAGEN spin-column kits. Do not use spin columns manufactured by other suppliers with the QIAcube Connect. Damage caused by use of other types of spin-columns will void the instrument warranty.
See: https://www.qiagen.com/media/product-tools/qiacube_connect-web/index.html for a demonstration on how the QIAcube is used for automated purification of nucleic acids.