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: 
If you say "I will need to take a shower", you mean "at some (understood) future time", which doesn't sound to me as if it matches your case, because the clause is supposed to be explaining why you can't come right now. 

I think you can concoct a scenario in which it would make sense ("I can't come right now, I will need to take a shower later so I have to do my other chores now"), but without some such it will sound odd. 

The fact that your shower might not be this very minute doesn't make any difference - the present tense is often used with future meaning in English (just as often as the so-called "future tense").
*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.