The channel attention mechanism adaptively recalibrates channel wise feature responses by modeling interdependencies between channels, an approach that has been successfully introduced in CNN architectures. However, this attention mechanism employs a scalar representation in the spatial domain for feature map channels, using the mean, which is a first-order statistic. Other works have utilized higher-order statistical information, such as variance and standard deviation, adding meaningful and informative aspects to the attention scalars. In this work, we extend the concept of using higher-order statistics to enhance the squeeze operation by focusing on the first four statistical orders of the feature map: mean, variance, skewness, and kurtosis. These can be encoded into a single entity using quaternion representation, and we modify the excitation block to process this quaternionic encoding using specialized linear layers that operate on quaternion data. We propose Quaternion Squeeze-and-Excitation Networks to enhance their ability to analyze and interpret complex datasets. We evaluate our model on several benchmark datasets, including ISIC 2016, ISIC 2017, ISIC 2018, KVASIR, ChestXray, and PH2. Experimental results demonstrate that the QuatSE U-net consistently outperforms traditional U-net and Attention U-net models in terms of Jaccard and Dice Scores.