I don't remember what happened to the first Chick and Duck that Joey and Chandler kept. Did they show in the show how they died? If yes, then please tell me the episode.
They died, probably of old age, but it wasn't shown in the show. It has been revealed in the final episode, when Joey bought the Chick Jr. and Duck Jr. The group had lied to Joey that they had gone to a special farm where they were very happy but could not be visited.
From the script:
It seems likely that Chandler had them humanely destroyed. In the final episode there's a mention that they were taken to a special farm that can't be visited in order to live out the remainder of their lives.
The average lifespan of a duck is 8-12 years and anything upwards of 20 years depending on the breed. The average lifespan of a well kept rooster can be anything between around 12-15 years so it seems highly unlikely either would be approaching old-age, even ignoring the possibility that they both coincidentally happened to die at the same time.
This whole gag is very strongly reminiscent of the 'Milner's Farm' joke from Season 1, episode 3 where Ross learns the fate of his beloved pet dog Chi-Chi.
The chick and the duck died of unknown reasons, likely age or eating something they shouldn't have. Joey was told they went to live on a farm because Joey is a giant man-child and can't be expected to handle the truth.
Rachel confirms they die in the last episode, without any stupid mention of being put down or euthanasia.
Chandler specifically kept the chick, and the duck, because he's against euthanasia. Instead of letting the chick be put down, he kept it, and ended up saving the duck, which was already mature/adult (not a duckling). He would never have them put down short of terminal illness or crippling age. He also wouldn't let Monica kill them, not that Monica would ever do that either. Neither of their characters are that heartless.
From Season 3 Episode 21, where the chick and the duck first appeared:
The Duck is some age older than the Chick, as it was fully grown by the time Chandler saved it. This can explain how they could have died near the same time.
They were last seen in Season 6, and the last time they were mentioned but not seen on screen before the final was Season 7 Episode 2 "The One With Rachel’s Book":