Can we calculate Z-score for any distribution? Is z-score only confined to normal distribution or can it be used for any distribution 
 A: Not always, but most of the time: In order to calculate a z-score for a sample, you need to have a non-zero sample variance, which requires that at least one of the values is different to the others.  Given a sample $x_1,...,x_n$ from a distribution, so long as you have a non-zero sample variance, the z-score for each sample value is:
$$z_i = \frac{x_i - \bar{x}_n}{s_n}.$$
For some distributions there is a non-zero probability that you will get all sample values being equal, which would give you a sample variance of zero.  If you use a point-mass distribution then this occurs almost surely, so it is even worse!  In the case where all sample values are equal it is not possible to obtain z-scores for the values.
(In this answer, I am assuming that when you refer to the z-score you are referring to standardisation using the sample, not the true moments of the distribution.  If you are referring to a z-score standardised by the true moments then the same basic principle applies --- i.e., you can only get a z-score if the distribution has a non-zero, non-infinite variance.)
A: The z-score is a variable that is centered and reduced ie
X your variable according to the distribution
$\mu$ your mean and V your deviance
$$Z = \frac{(X - \mu)}{\sqrt{V}}$$
You can use estimators for both $\mu$ and V
So yes, it can be used for any distribution as long as you have both the mean and deviance
Note : In case V = 0, Z is not well defined.
