Is a cold shower a healthier thing than a hot shower?

Extremes : of temperature may not be healthy (cold or hot). However, if you ever live in a place like Finland you may people go into a hot sauna and then jump into a freezing river. Hot to cold repeatedly can be used as a form of hydrotherapy. I am not sure that one can say if hot or cold is better. But marked extremes are not always well tolerated by the body.

No.: No preferential health benefit for one over the other. Might be reasonable to avoid both if you have a serious cardiac or blood pressure condition, but I think it would have to be quite serious and the shower quite extreme to produce any symptoms.