Question: Can we use "I can't come right now, I need to take a shower" if we are informing a friend? Or should it be "I can't come right now, I will need to take a shower"?

Answer: 
"I can't come right now, I need to take a shower."

This means that because you have something else to do, namely taking a shower, you can't come right now. Note that it implies that even if you aren't going, you still need to take a shower.

"I can't come right now, I will need to take a shower."

This means that you don't necessarily need to take a shower now. However, if you were to come, you couldn't do it now because you would have to take a shower first. This does not imply that you need to take a shower unless you are going to come.

"I need" is a present need. "I will need" is a conditional need at an unspecified future time.
*I need to take a shower* is the correct variant in the situation you describe. The present tense is used because the **need** exists in the present. 

In the following examples the future tense is used because the need will not exist until some time in the future:

* If I decide to go out after all, I will need to take a shower.
* As soon as I arrive I will need to find somewhere to eat.