He never said anything bad about Armenians.
Here is the line that you mean:
Поди ты прочь – ты мне не сын,
Ты не чеченец – ты старуха,
Ты трус, ты раб, ты армянин.
In English approximately:
Go away - you're not my son any more
you're not a chechen, you're an old lady
you're coward, you're a slave, you're Armenian.
Now, in the poem, these words are pronounced my an old chechen man, who is blaming his son for NOT killing and NOT stealing. In his understanding, the man, who doesn't kill and doesn't steal is coward, is a slave, and is ARMENIAN.
The son couldn't kill the enemy because he was:
Убийца был
Один, изранен, безоружен...
English: the enemy was alone, wounded, armless.
The only thing Pushkin wanted to say is that chechens were of opinion that an Armenian would never kill a defendless person.So, again, we should thank the great Pushkin for his being fair.
Any other interpretation given out of context, and manipulated by azers, is totally wrong.