In the recent search for additional post-quantum designs, multivariate quadratic equations (MQE) based designs have been receiving attention due to their small signature sizes. Unbalanced Oil and Vinegar (UOV) is an MQE-based digital signature (DS) scheme proposed over two decades ago. Although the mathematical security of UOV has been thoroughly analyzed, several practical side-channel attacks (SCA) have been shown on UOV based DS schemes. In this work, we perform a thorough analysis to identify the variables in UOV based DS schemes that can be exploited with passive SCA, specifically differential power attacks (DPA). Secondly, we introduce masking as a countermeasure to protect the sensitive components of UOV based schemes. We propose efficient masked gadgets for all the critical operations, including the masked dot-product and matrix-vector multiplication. We show that our gadgets are secure in the t-probing model through formal proofs, mechanically verified using the maskVerif tool. We implemented and demonstrated the practical feasibility of our arbitrary-order masking algorithms for UOV-Ip and UOV-III. We show that the masked signature generation of UOV-Ip performs up to 62% better than ML-DSA-44 and 99% better than Falcon-512. In addition, the security of our implementation is practically validated using the test vector leakage assessment (TVLA) methodology.