Q. If I work remotely from Florida for a new Jersey company, do I owe New Jersey state tax?
A. No.
You said "I filled a non resident new Jersey return and owed taxes".
You should not have filed a NJ return, unless you actually worked in N.J. (e.g. trip to your employer's office in NJ). Since you did (apparently) erroneously prepare an NJ return, it would be easy for TurboTax (TT) to mistakenly calculate NJ tax. In the personal info section, you, most likely, answered yes to the question "did you make money in another state". You should have answered no (unless you did spend some time at your employer's office in NJ).
You most likely just need to file an amended NJ return to claim a refund of what your previously paid. As @pk suggested, you may want to contact NJDOR for specific advise.
