The properties of galaxy clusters as a function of redshift can be utilized as an important cosmological tool. We present initial results from a program of follow-up observations of the Sunyaev-Zeldovich effect (SZE) in high-redshift galaxy clusters detected by the Massive and Distant Clusters of WISE Survey (MaDCoWS) which uses infrared data from the Wide-field Infrared Survey (WISE) instrument. Using typical on-source integration times of 3-4 hr per cluster, MUSTANG2 on the Green Bank Telescope was able to measure strong detections of SZE decrements and statistically significant masses on 14 out of 16 targets. On the remaining two, weaker (3.7 ) detections of the SZE signal and strong upper limits on the masses were obtained. In this paper we present masses and pressure profiles of each target and outline the data analysis used to recover these quantities. Of the clusters with strong detections, three show significantly flatter pressure profiles while, from the MUSTANG2 data, five others show signs of disruption at their cores. However, outside of the cores of the clusters, we were unable to detect significant amounts of asymmetry. Finally, there are indications that the relationship between optical richness used by MaDCoWS and SZE-inferred mass may be significantly flatter than indicated in previous studies.