The image of glass in the Glass Menagerie is an allegory to characteristics that are specific to Laura: Fragility and purity. Fragility means that she is very easy to shatter, break, and fall apart, like glass. Moreover, glass is clear just as she is innocent,and pure. These two latter descriptors make Laura both as rare and breakable as the broken unicorn in her glass menagerie.
Similarly in Cinderella, the glass slipper is as unique as Cinderella's situation: She is an innocent, pure woman whose evil stepmother and stepsisters try to "break" consistently. Her freedom is easily fragile, as it depends entirely of the success of one night. It is interesting, however, that the fact that the shoe represents her fragility and purity was the precise memento that she left behind for the Prince- and the one he cherished the most. It is almost as if the shoe represents the qualities a man as good as the Prince should look for in his future bride.
In Snow White, her innocence, fragility, and purity are also shown, but this time they can be clearly appreciated behind the glass coffin, where she lies as a victim of an unfair queen. Her salvation is having her first kiss from the Prince. Once again we find the idea of the Prince saving the damsel in distress, just like in Cinderella and in The Glass Menagerie.
Clearly, Laura was at a loss because her Prince Charming came to her at the wrong time. Hence, her innocence and purity will go on forever untouched. Allegorically, when the unicorn broke its horn was indicative of her potential being lost forever.