When you negate the sentence, the thing being possessed needs to be in the genitive case.
У меня есть яблоко → У меня нет яблока
Спасибо. I was obviously uncertain, but had already answered with another excepted string of words.
Accepted Not much difference in pronunciation, just to confuse people :-)
Technically их is the genitive form but when using preposition у, the н is added in front of их because it is difficult to pronounce the у and и sounds one after the other.