No. Just because you can say a thing, does not imply the thing has an identifiable referent. Noam Chomsky famously argued this with the example sentence "colorless green ideas sleep furiously", which is grammatically correct, despite no one on Earth including himself being able to tell you how something can be simultaneously green and colorless, what it even means for an idea to be colored in the first place, or how an immaterial concept like "idea" can sleep, let alone sleep "furiously". No part of the sentence comes attached to an identifiable concept - yet, it is a syntactically valid sentence, all the same.
The existence of the phrase "I love a round square" does not prove that round squares are a thing, only that that phrase itself does not violate the syntactic constraints of English.