@oristarA,
Good grief, no. Get rid of that dictionary. Hip=depression??? That's a bizarre definition.
'Hip', as it is generally understood by most English-speakers,especially the younger generation, means to be sophisticated, knowledgeable,
au courant, 'with it', in the mainstream etc. etc. This meaning of the word derives primarily from the 'hippies', the very influential youth sub-culture which prevailed in the USA in the 1960s and '70s. Prior to that era, the word was usually rendered as 'hep,' a slang term of jazz musicians, meaning essentially the same thing. To be 'square' was the direct opposite of being 'hip' or 'hep.' Modern-day equivalents might be 'nerdy' or 'dweeb'.
But the expression you quote -- "Be there or be square" -- was for a while a catch-phrase, popular mainly because 'there' and 'square' rhyme with each other. The expressin is still in sporadic use.