John uses the Shakespearean line, "eternity was in our lips and eyes," when he is in the middle of a conversation with Dr. Shaw, who is looking after John's mother, Linda. Linda, upon her return to civilization, has taken so much soma that she is constantly "absent," and gone from any interactions. They let her have as much as she wants, because she is old, she "hasn't got any serious work" to do, as Dr. Shaw puts it, and because she just wants to escape. She has lived a hard life away from it, and was so preconditioned to her previous life that she never really adjusted. So upon her return, she just escapes.
John feels that giving her so much is detrimental to her health. He doesn't understand why they would let her shorten her life so much just to take soma. The doctor explains that yes, it will shorten her life by several years, but that on a "soma-holiday" she will experience "a bit of what our ancestors used to call eternity." He is meaning that in her dreams, she experiences years and years of happiness, adventures, and journeys. It seems like forever to her.
At this point, John understands a bit. The books states, "John began to understand," and then he quotes the Shakespeare line you mentioned above. I'm not sure that John really meant anything by it, other than the fact that he understood what "eternity" meant, because of his exposure to Shakespeare, and that according to Shakespeare, eternity wasn't a unit of time, a literal expansion of time, but rather something that we hold within ourselves, or, in our "lips and eyes." John was saying that he understood how Linda could be experiencing a figurative eternity, because Shakespeare himself had said that our own minds control our concept of eternity; our minds are capable of helping us to feel it and experience it.
Shakespeare helped John understand a lot of things in civilization, and the concept of dreaming, taking a break through the power of your mind, or a figurative "eternity" was one of them. It helped him to understand why Linda was so content to just escape into "eternity" in her mind. I hope that those thoughts help a bit; good luck!
I believe that this quote refers to John's sudden realization that eternity, or in this case percieved hapiness, is what we see and taste but not what may actually exist.