A souffle shouldn't taste strongly of eggs as one of the key components of a good souffle is an intensely flavoured base.
As for the mushyness it depends on what else is in the souffle as that would impact on how dry or mushy the end result would be. However if it is mushy as in uncooked eggs mushy then no it shouldn't be like that.
My solution for the eggy-taste would be to ensure that the egg whites are thoroughly folded (though not so much to deflate them) through the base mixture as it could be that you hit a pocket of unflavoured egg. In order to mix the mushyness I would look at how much liquid the base has as well as perhaps leaving the souffles in just that little bit longer to ensure that they are cooked through (also eggwhites that have deflated too much could also cause this as the souffle would be denser, requiring a longer cooking time).
Hope this helps.