They're both acceptable. "Take a shower" is more common in American English.

I think that either one is acceptable. To tell you the truth, I am more likely to use 'take a shower' if it's necessary to choose between them. However, I usually just use shower as a verb. Like: "After I finish my homework, I'm going to shower and then go to bed."

Like many phrases involving delexical verbs, you have a choice of 'take' or 'have'. Both are acceptable, with the 'have' version being more common in Britain, and the 'take' version being more common in the US.

And likewise 'have a shower' is more common in British English, or at least more common in the UK. Either is correct though.