Details are at the MLA website:
Begin the entry in the works-cited list with the author’s real name and, in parentheses, user name, if both are known and they differ. If only the user name is known, give it alone.Here is a helpful graphic:
Next provide the entire text of the tweet in quotation marks, without changing the capitalization. Conclude the entry with the date and time of the message and the medium of publication (Tweet). For example:
Athar, Sohaib (ReallyVirtual). "Helicopter hovering above Abbottabad at 1AM (is a rare event)." 1 May 2011, 3:58 p.m. Tweet.