One approach is to use soft clustering methods to find the clusters and then check to see if they are gaussian. If they are, you can use a GMM model to represent the entire dataset.
For more: https://stats.stackexchange.com/questions/260116/when-to-use-gaussian-mixture-model
GMM is a kind of cluster algorithm. If you know the number of clusters and the corresponding proportion of clusters, GMM can be implemented.
It is no need to test whether the data is for GMM. If you do want to test,try to use Kmeans or whatever you want and then check if the data is subjected to Gaussian distribution.
Know I get the answer, R-squared can not be used to check the goodness of GMM. R-squared works for independent and dependent variables and we need to measure the percentage that represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables