Can MHC class I be used for presenting peptides of extracellular origin by non-professional APCs?
when non-professional APCs that can undergo phagocytosis (Eosinophil, neutrophil, dendritic cell) engulf bacteria


Dendritic cells and probably neutrophils are professional APC, so this part of the question is a little confusing.

You're asking about cross-presentation, which is the process by which extracellular antigen can be presented to CD8 T cells on MHC class I.  Cross-presentation is mainly performed by certain sub-classes of dendritic cell. A number of other cell types have been shown to be capable of it, but it's not clear how important they are in vivo.

A decent review of cross-presentation is Understanding the Biology of Antigen Cross-Presentation for the Design of Vaccines Against Cancer.  A review of a related but somewhat different concept (MHC recycling) is Endocytic Recycling of MHC Class I Molecules in Non-professional Antigen Presenting and Dendritic Cells.