Welcome to the forum. I'll try to help. But in judging this, it would be helpful to know about your risk of STD exposure. Can you say something about your sexual lifestyle and relationships, recent new partners, sex or partners, and so on? Also, were tests for gonorrhea and chlamydia not done before you started treatment?
Based on the information provided so far, I doubt you have gonorrhea, which usually causes more prominent urethral discharge than you have described. Instead of squeezing out a very small amount, gonorrhea usually causes profuse yellow pus dripping from the penis. There are exceptions, but they aren't common.
The combination of urethritis with conjunctivitis conceivably could indicate chlamydia, but minocycline is always effective against chlamydia -- so persistent urethral symptoms and onset of eye symptoms while on that treatment rules out chlamydia as a cause.
I think there's a good chance you are infected with adenovirus, which causes around 3-5% of nongonococcal urethritis (NGU) cases. Adeno is mostly a cause of respiratory infections (colds, bronchitis, and sometimes pneumonia), with or without conjunctivitis, but some strains can cause urethritis. Adeno might be even more likely if you had received oral sex in the week or so before your symptoms began. Lack of response to treatment fits with adeno, since no viruses respond to any antibiotics.
Here are three threads that go into adenovirus and NGU. You can find others by using the search link.
http://www.medhelp.org/posts/STDs/Possible-Adenovirus-Urethritis-Conjunctivitis/show/705605
http://www.medhelp.org/posts/STDs/Urethritis-Herpes-or-Other-Virus/show/1113305
http://www.medhelp.org/posts/STDs/Painful-Urination--Antibiotics-havent-worked/show/1041584
Many (most?) doctors are unaware of the adenovirus/NGU connection, and yours might be among them. He might appreciate being brought up to speed. If that's what you have, you can expect your symptoms -- both eye and urethral -- to clear up on their own after week or two. No long term harm will come.
Let me know the missing information about testing and your sexual exposures and perhaps I'll have more to advise. In the meantime, I wouldn't be very worried. This isn't likely anything serious.
Regards-- HHH, MD