I'm posting this because of the impact that losing a pet can have on a child. Pets just can't "disappear" only to have us adults explain "what happened". Death is a mystery. It can't be explained. This is a critical moment in your child learning about death and the expression of loss.
Depending on whether or not attending the euthanizing process will benefit your child, it may be a way to "concretize" the event meaning that your child witnesses it and does not have to imagine it. The peacefulness of the final moments may bring them peace, as well. If not, they may have very sad thoughts over things that didn't happen or how their dog or cat was feeling. Kids may also not be able to explain what they are feeling or thinking.
The death of their pet, no matter what kind of animal, is a chance to teach them about death in ways they can understand. They can see you grieve in a healthy fashion, yes even if you are a blubbering mess, and learn how to continue to love through memories. A life well-lived has a lot of loss, doesn't it?
The grief process is "allowed" to go on. Talking about our pet even weeks later and feeling emotional is typical. Let your kids see that your pet is not forgotten.
Think of a way to honor your pet. Take old blankets and food to a shelter or a rescue organization in their honor. Avoid just writing a check. Once again, "concretize" the process. Actively engaging in making another animal's life better is healthy.
At times, the grief spreads to "anticipatory grief" of losing a parent. Most parents don't die before their children, but unfortunately, many do. If your child expresses grief over you dying, you may want to consider this approach. It worked for us when our son was 7 and his friend's mother died of cancer at age 37. All of us were very close.
My husband and I opened up a big paper bag so that it was flat. On the extreme left side was an "X" for "the time we are now" and we drew a line to the extreme right side, marking the time we were most likely to die.
We explained to our son that by the time we die, he'll be 50 years old with a teenage son. He was 7 and being a teenager sounded like light years away. It gave him perspective and he was immediately calm and reassured and never expressed concerns again.
Just do the best you can.