Oh, sorry! I thought you were talking about the tank water. OK - I'm looking at the MSDS for the API Test Kit now....
Yes! It will make the ammonia test vial cloudy!
Ammonia Test Solution #2 contains sodium hydroxide. This reacts with the Epsom salt, forming some magnesium hydroxide. This has a low solubility in water, which would result in cloudiness.
Not sure about the false negative, but it's definitely a potential issue since the side reaction (formation of magnesium hydroxide) is occurring, which means there's less hydroxide available for the test itself.
You may want to try adding an extra drop of Ammonia Solution #2 to the vial to counter the side reaction. (Solution #1 is OK. It's only #2 that has the NaOH.)
Also, you may want to contact API to ask about this. They could probably tell you whether a false negative is possible. (And whether you need the extra drop of Solution #2, since using it up faster means you'd reduce the number of times you could test.)
Haven't looked at the solutions for the nitrate or nitrite tests yet, but it's 3am, and my brain is oatmeal.

Will look at those tomorrow, if you'd like.
API Freshwater Master Test Kit MSDS:
http://cms.marsfishcare.com/files/ms...est_122309.pdf