Shakespeare is not good english. His work is old, and the language has evolved, and his work is poetry.
Also, citation needed for using 'they' as singular. Apart from the fact that it very strongly sounds plural, one of my personal pet peeves with English is the fact that we do not have a singular plural differentiation in the second person. As long as we are re-working pronouns, lets not repeat that mistake.
When I am trying to be careful and gender neutral (for something that needs to be speakable, so no (s)he), I try to structure it to use either 'one' or 'that person' in place of (s)he.