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Information about the structure and composition of biopsy specimens can assist in disease monitoring and diagnosis. In principle, this can be acquired from Raman and infrared (IR) hyperspectral images (HSIs) that encode information about how a sample's constituent molecules are arranged in space. Each tissue section/component is defined by a unique combination of spatial and spectral features, but given the high dimensionality of HSI datasets, extracting and utilising them to segment images is non-trivial. Here, we show how networks based on deep convolutional autoencoders (CAEs) can perform this task in an end-to-end fashion by first detecting and compressing relevant features from patches of the HSI into low-dimensional latent vectors, and then performing a clustering step that groups patches containing similar spatio-spectral features together. We showcase the advantages of using this end-to-end spatio-spectral segmentation approach compared to i) the same spatio-spectral technique not trained in an end-to-end manner, and ii) a method that only utilises spectral features (spectral k-means) using simulated HSIs of porcine tissue as test examples. Secondly, we describe the potential advantages/limitations of using three different CAE architectures: a generic 2D CAE, a generic 3D CAE, and a 2D CNN architecture inspired by the recently proposed UwU-net that is specialised for extracting features from HSI data. We assess their performance on IR HSIs of real colon samples. We find that all architectures are capable of producing segmentations that show good correspondence with HE stained adjacent tissue slices used as approximate ground truths, indicating the robustness of the CAE-driven approach for segmenting biomedical HSI data. Additionally, we stress the need for more accurate ground truth information to rigorously compare the advantages offered by each architecture.